{"id":678,"date":"2021-08-13T16:36:26","date_gmt":"2021-08-13T16:36:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&p=678"},"modified":"2023-12-05T09:01:44","modified_gmt":"2023-12-05T09:01:44","slug":"independent-and-mutually-exclusive-events-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/independent-and-mutually-exclusive-events-2\/","title":{"raw":"Mutually Exclusive Events","rendered":"Mutually Exclusive Events"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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    \r\n \t
  • Determine if two events are mutually exclusive<\/li>\r\n \t
  • Calculate probabilities for events that are mutually exclusive and events that are not mutually exclusive<\/li>\r\n \t
  • Calculate probabilities to determine if two events are independent<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n

    Mutually Exclusive Events<\/h2>\r\n[latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive<\/strong> events if they cannot occur at the same time. This means that [latex]A[\/latex] and [latex]B[\/latex] do not share any outcomes and [latex]P(A \\text{ AND } B) = 0[\/latex].\r\n\r\nFor example, suppose the sample space\u00a0[latex]S[\/latex] = {[latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[\/latex]}. Let [latex]A[\/latex] = {[latex]1, 2, 3, 4, 5[\/latex]}, [latex]B[\/latex] = {[latex]4, 5, 6, 7, 8[\/latex]}, and [latex]C[\/latex] = {[latex]7, 9[\/latex]}. [latex]A \\text{ AND } B[\/latex] = {[latex]4, 5[\/latex]}. [latex]\\displaystyle{P}{({A} \\text{ AND } {B})}=\\frac{{2}}{{10}}[\/latex]and is not equal to zero. Therefore, [latex]A[\/latex] and [latex]B[\/latex] are not mutually exclusive. [latex]A[\/latex] and [latex]C[\/latex] do not have any numbers in common so [latex]P(A \\text{ AND } C) = 0[\/latex]. Therefore, [latex]A[\/latex] and [latex]C[\/latex] are mutually exclusive.\r\n\r\nIf it is not known whether\u00a0[latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, assume they are not until you can show otherwise<\/strong>. The following examples illustrate these definitions and terms.\r\n

    <\/h3>\r\n
    \r\n

    Example<\/h3>\r\nFlip two fair coins. (This is an experiment.)\r\n\r\nThe sample space is {[latex]HH[\/latex], [latex]HT[\/latex], [latex]TH[\/latex], [latex]TT[\/latex]} where [latex]T[\/latex] = tails and [latex]H[\/latex] = heads. The outcomes are [latex]HH[\/latex], [latex]HT[\/latex], [latex]TH[\/latex], and [latex]TT[\/latex]. The outcomes HT and TH are different. The [latex]HT[\/latex] means that the first coin showed heads and the second coin showed tails. The [latex]TH[\/latex] means that the first coin showed tails and the second coin showed heads.\r\n
      \r\n \t
    • Let [latex]A[\/latex] = the event of getting at most one tail<\/strong>. (At most one tail means zero or one tail.) Then [latex]A[\/latex] can be written as {[latex]HH[\/latex], [latex]HT[\/latex], [latex]TH[\/latex]}. The outcome [latex]HH[\/latex] shows zero tails. [latex]HT[\/latex] and [latex]TH[\/latex] each show one tail.<\/li>\r\n \t
    • Let [latex]B[\/latex] = the event of getting all tails. [latex]B[\/latex] can be written as {[latex]TT[\/latex]}. [latex]B[\/latex] is the complement<\/strong> of [latex]A[\/latex], so [latex]B = A'[\/latex]. Also, [latex]P(A) + P(B)[\/latex] = [latex]P(A) + P(A') = 1[\/latex].<\/li>\r\n \t
    • The probabilities for [latex]A[\/latex] and for [latex]B[\/latex] are [latex]P(A) = \\displaystyle\\frac{{3}}{{4}}[\/latex] and [latex]P(B) = \\displaystyle\\frac{{1}}{{4}}[\/latex].<\/li>\r\n \t
    • Let [latex]C[\/latex] = the event of getting all heads. [latex]C[\/latex] = {[latex]HH[\/latex]}. Since [latex]B = {TT}[\/latex], [latex]P(B \\text{ AND } C) = 0[\/latex]. [latex]B[\/latex] and [latex]C[\/latex] are mutually exclusive. [latex]{(}B[\/latex] and [latex]C[\/latex] have no members in common because you cannot have all tails and all heads at the same time.)<\/li>\r\n \t
    • Let [latex]D[\/latex] = event of getting more than one<\/strong> tail. [latex]D = {TT}[\/latex]. [latex]P(D) = \\displaystyle\\frac{{1}}{{4}}[\/latex]<\/li>\r\n \t
    • Let [latex]E[\/latex] = event of getting a head on the first roll. (This implies you can get either a head or tail on the second roll.) [latex]E = {HT,HH}[\/latex]. [latex]P(E) = \\displaystyle\\frac{{2}}{{4}}[\/latex]<\/li>\r\n \t
    • Find the probability of getting at least one<\/strong> (one or two) tail in two flips. Let [latex]F[\/latex] = event of getting at least one tail in two flips. [latex]F = {HT, TH, TT}[\/latex]. [latex]P(F) = \\displaystyle\\frac{{3}}{{4}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

      <\/h3>\r\n
      \r\n

      try it<\/h3>\r\nDraw two cards from a standard [latex]52[\/latex]-card deck with replacement. Find the probability of getting at least one black card.\r\n\r\n[reveal-answer q=\"124078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124078\"]\r\n\r\nThe sample space of drawing two cards with replacement from a standard 52-card deck with respect to color is {[latex]BB[\/latex], [latex]BR[\/latex], [latex]RB[\/latex], [latex]RR[\/latex]}.\r\n\r\nEvent\u00a0[latex]A[\/latex] = Getting at least one black card = {[latex]BB[\/latex], [latex]BR[\/latex], [latex]RB[\/latex]}\r\n\r\n[latex]\\displaystyle{P}{({A})}=\\frac{{3}}{{4}}={0.75}[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

      <\/h3>\r\n
      \r\n

      Example<\/h3>\r\nFlip two fair coins. Find the probabilities of the events.\r\n
        \r\n \t
      1. Let [latex]F[\/latex] = the event of getting at most one tail (zero or one tail).<\/li>\r\n \t
      2. Let [latex]G[\/latex] = the event of getting two faces that are the same.<\/li>\r\n \t
      3. Let [latex]H[\/latex] = the event of getting a head on the first flip followed by a head or tail on the second flip.<\/li>\r\n \t
      4. Are [latex]F[\/latex] and [latex]G[\/latex] mutually exclusive?<\/li>\r\n \t
      5. Let [latex]J[\/latex] = the event of getting all tails. Are [latex]J[\/latex] and [latex]H[\/latex] mutually exclusive?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124079\"]\r\n\r\nSolution:\r\n\r\nLook at the sample space in Example 3.\r\n
          \r\n \t
        1. Zero ([latex]0[\/latex]) or one ([latex]1[\/latex]) tails occur when the outcomes [latex]HH[\/latex], [latex]TH[\/latex], [latex]HT[\/latex] show up. [latex]P(F) = \\displaystyle\\frac{{3}}{{4}}[\/latex]<\/li>\r\n \t
        2. Two faces are the same if [latex]HH[\/latex] or [latex]TT[\/latex] show up. [latex]P(G) = \\displaystyle\\frac{{2}}{{4}}[\/latex]<\/li>\r\n \t
        3. A head on the first flip followed by a head or tail on the second flip occurs when [latex]HH[\/latex] or [latex]HT[\/latex] show up. [latex]P(H) = \\displaystyle\\frac{{2}}{{4}}[\/latex]<\/li>\r\n \t
        4. [latex]F[\/latex] and [latex]G[\/latex] share [latex]HH[\/latex] so [latex]P(F \\text{ AND } G)[\/latex] is not equal to zero ([latex]0[\/latex]). [latex]F[\/latex] and [latex]G[\/latex] are not mutually exclusive.<\/li>\r\n \t
        5. Getting all tails occurs when tails shows up on both coins ([latex]TT[\/latex]). [latex]H[\/latex]'s outcomes are [latex]HH[\/latex] and [latex]HT[\/latex]. [latex]J[\/latex] and [latex]H[\/latex] have nothing in common so [latex]P(J \\text{ AND } H) = 0[\/latex]. [latex]J[\/latex] and [latex]H[\/latex] are mutually exclusive.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\n
          \r\n\r\nThis video provides two more examples of finding the probability of events that are mutually exclusive.\r\n\r\nhttps:\/\/youtu.be\/zxhDOvS2c3k\r\n

          <\/h4>\r\n
          \r\n

          try it<\/h3>\r\nA box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Find the probability of the following events:\r\n
            \r\n \t
          1. Let [latex]F[\/latex] = the event of getting the white ball twice.<\/li>\r\n \t
          2. Let [latex]G[\/latex] = the event of getting two balls of different colors.<\/li>\r\n \t
          3. Let [latex]H[\/latex] = the event of getting white on the first pick.<\/li>\r\n \t
          4. Are [latex]F[\/latex] and [latex]G[\/latex] mutually exclusive?<\/li>\r\n \t
          5. Are [latex]G[\/latex] and [latex]H[\/latex] mutually exclusive?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124080\"]\r\n
              \r\n \t
            1. [latex]P(F) = \\displaystyle\\frac{{1}}{{4}}[\/latex]<\/li>\r\n \t
            2. [latex]P(G) = \\displaystyle\\frac{{1}}{{2}}[\/latex]<\/li>\r\n \t
            3. [latex]P(H) = \\displaystyle\\frac{{1}}{{2}}[\/latex]<\/li>\r\n \t
            4. Yes<\/li>\r\n \t
            5. No<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

              <\/h3>\r\n
              \r\n

              Example<\/h3>\r\nRoll one fair, six-sided die. The sample space is {[latex]1, 2, 3, 4, 5, 6[\/latex]}. Let event\r\n[latex]A[\/latex] = a face is odd. Then [latex]A[\/latex] = {[latex]1, 3, 5[\/latex]}. Let event [latex]B[\/latex] = a face is even. Then [latex]B[\/latex] = {[latex]2, 4, 6[\/latex]}.\r\n
                \r\n \t
              • Find the complement of [latex]A[\/latex], [latex]A'[\/latex]. The complement of [latex]A[\/latex], [latex]A'[\/latex], is [latex]B[\/latex] because [latex]A[\/latex] and [latex]B[\/latex] together make up the sample space. [latex]P(A) + P(B) = P(A) + P(A') = 1[\/latex]. Also, [latex]P(A) = \\displaystyle\\frac{{3}}{{6}}[\/latex] and [latex]P(B) = \\displaystyle\\frac{{3}}{{6}}[\/latex].<\/li>\r\n \t
              • Let event [latex]C[\/latex] = odd faces larger than two. Then [latex]C[\/latex] = {[latex]3, 5[\/latex]}. Let event [latex]D[\/latex] = all even faces smaller than five. Then [latex]D[\/latex] = {[latex]2, 4[\/latex]}. [latex]P(C \\text{ AND } D) = 0[\/latex] because you cannot have an odd and even face at the same time. Therefore, [latex]C[\/latex] and [latex]D[\/latex] are mutually exclusive events.<\/li>\r\n \t
              • Let event [latex]E[\/latex] = all faces less than five. [latex]E[\/latex] = {[latex]1, 2, 3, 4[\/latex]}.<\/li>\r\n<\/ul>\r\nAre\u00a0[latex]C[\/latex] and [latex]E[\/latex] mutually exclusive events? (Answer yes or no.) Why or why not?\r\n[reveal-answer q=\"124081\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124081\"]\r\nSolution:\r\n\r\nNo.\r\n[latex]C[\/latex] = {[latex]3, 5[\/latex]} and [latex]E[\/latex] = {[latex]1, 2, 3, 4[\/latex]}. [latex]P(C \\text{ AND } E) = \\displaystyle\\frac{{1}}{{6}}[\/latex]. To be mutually exclusive, [latex]P(C \\text{ AND } E)[\/latex] must be zero.\r\n[\/hidden-answer]\r\n
                  \r\n \t
                • Find [latex]P(C|A)[\/latex]. This is a conditional probability. Recall that the event [latex]C[\/latex] is {[latex]3, 5[\/latex]} and event [latex]A[\/latex] is {[latex]1, 3, 5[\/latex]}. To find [latex]P(C|A)[\/latex], find the probability of [latex]C[\/latex] using the sample space [latex]A[\/latex]. You have reduced the sample space from the original sample space {[latex]1, 2, 3, 4, 5, 6[\/latex]} to {[latex]1, 3, 5[\/latex]}. So, [latex]P(C|A) = \\displaystyle\\frac{{2}}{{3}}[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n
                  \r\n

                  try it<\/h3>\r\nLet event\u00a0[latex]A[\/latex] = learning Spanish. Let event [latex]B[\/latex] = learning German. Then [latex]A[\/latex] AND [latex]B[\/latex] = learning Spanish and German. Suppose [latex]P(A) = 0.4[\/latex] and [latex]P(B) = 0.2[\/latex]. [latex]P(A \\text{ AND } B) = 0.08[\/latex]. Are events [latex]A[\/latex] and [latex]B[\/latex] independent? Hint: You must show ONE of the following:\r\n
                    \r\n \t
                  • [latex]P(A|B) = P(A)[\/latex]<\/li>\r\n \t
                  • [latex]P(B|A)[\/latex]<\/li>\r\n \t
                  • [latex]P(A \\text{ AND } B) = P(A)P(B)[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"124082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124082\"]\r\n[latex]P(A|B)=\\displaystyle\\frac{{{P}{({A}\\text{ AND }{B})}}}{{{P}{({B})}}}=\\frac{{0.08}}{{0.2}}={0.4}={P}{({A})}[\/latex]\r\n\r\nThe events are independent because\u00a0[latex]P(A|B) = P(A)[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
                    \r\n

                    Recall: Multiplying and Dividing Decimals<\/h3>\r\n

                    Multiply Decimal Numbers<\/h4>\r\n
                      \r\n \t
                    1. Determine the sign of the product.<\/li>\r\n \t
                    2. Write the numbers in vertical format, lining up the numbers on the right.<\/li>\r\n \t
                    3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\r\n \t
                    4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\r\n \t
                    5. Write the product with the appropriate sign.<\/li>\r\n<\/ol>\r\n

                      Divide Decimal Numbers<\/h4>\r\n
                        \r\n \t
                      1. Determine the sign of the quotient.<\/li>\r\n \t
                      2. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.<\/li>\r\n \t
                      3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\r\n \t
                      4. Write the quotient with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                        \r\n

                        Example<\/h3>\r\nLet event\u00a0[latex]G[\/latex] = taking a math class. Let event [latex]H[\/latex] = taking a science class. Then, [latex]G[\/latex] AND [latex]H[\/latex] = taking a math class and a science class. Suppose [latex]P(G) = 0.6[\/latex], [latex]P(H) = 0.5[\/latex], and [latex]P(G \\text{ AND } H) = 0.3[\/latex]. Are [latex]G[\/latex] and [latex]H[\/latex] independent?\r\n\r\nIf\u00a0[latex]G[\/latex] and [latex]H[\/latex] are independent, then you must show ONE<\/strong> of the following:\r\n
                          \r\n \t
                        • [latex]P(G|H) = P(G)[\/latex]<\/li>\r\n \t
                        • [latex]P(H|G) = P(H)[\/latex]<\/li>\r\n \t
                        • [latex]P(G \\text{ AND } H) = P(G)P(H)[\/latex]<\/li>\r\n<\/ul>\r\nNote:\r\n\r\nThe choice you make depends on the information you have.<\/strong> You could choose any of the methods here because you have the necessary information.\r\n
                            \r\n \t
                          1. Show that [latex]P(G|H) = P(G)[\/latex].<\/li>\r\n \t
                          2. Show [latex]P(G \\text{ AND } H) = P(G)P(H)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124083\"]\r\nSolution:\r\n
                              \r\n \t
                            1. [latex]P(G|H) = \\frac{P(G \\text{ AND } H)}{P(H)} = \\frac{0.3}{0.5} = 0.6 = P(G)[\/latex]<\/li>\r\n \t
                            2. [latex]P(G)P(H) = (0.6)(0.5) = 0.3 = P(G \\text{ AND } H)[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\nSince\u00a0[latex]G[\/latex] and [latex]H[\/latex] are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent (that is, they are dependent) then knowing that a person is taking a science class would change the chance he or she is taking math. For practice, show that [latex]P(H|G) = P(H)[\/latex] to show that [latex]G[\/latex] and [latex]H[\/latex] are independent events.\r\n\r\n<\/div>\r\n \r\n
                              \r\n

                              try it<\/h3>\r\nIn a bag, there are six red marbles and four green marbles. The red marbles are marked with the numbers [latex]1, 2, 3, 4, 5[\/latex], and [latex]6[\/latex]. The green marbles are marked with the numbers [latex]1, 2, 3[\/latex], and [latex]4[\/latex].\r\n
                                \r\n \t
                              • [latex]R[\/latex] = a red marble<\/li>\r\n \t
                              • [latex]G[\/latex] = a green marble<\/li>\r\n \t
                              • [latex]O[\/latex] = an odd-numbered marble<\/li>\r\n \t
                              • The sample space is [latex]S[\/latex] = {[latex]R1, R2, R3, R4, R5, R6, G1, G2, G3, G4[\/latex]}.<\/li>\r\n<\/ul>\r\n[latex]S[\/latex] has ten outcomes. What is [latex]P(G \\text{ AND } O)[\/latex]?\r\n\r\n[reveal-answer q=\"124084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124084\"]\r\nEvent\u00a0[latex]G[\/latex] and [latex]O[\/latex] = {[latex]G1, G3[\/latex]}\r\n\r\n[latex]P(G \\text{ AND } O) = \\displaystyle\\frac{{2}}{{10}}[\/latex] = 0.2\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

                                <\/h3>\r\n
                                \r\n

                                Example<\/h3>\r\nLet event\u00a0[latex]C[\/latex] = taking an English class. Let event [latex]D[\/latex] = taking a speech class.\r\n\r\nSuppose\u00a0[latex]P(C) = 0.75[\/latex], [latex]P(D) = 0.3[\/latex], [latex]P(C|D) = 0.75[\/latex] and [latex]P(C \\text{ AND } D) = 0.225[\/latex].\r\n\r\nJustify your answers to the following questions numerically.\r\n
                                  \r\n \t
                                1. Are [latex]C[\/latex] and [latex]D[\/latex] independent?<\/li>\r\n \t
                                2. Are [latex]C[\/latex] and [latex]D[\/latex] mutually exclusive?<\/li>\r\n \t
                                3. What is [latex]P(D|C)[\/latex]?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124085\"]\r\nSolution:\r\n
                                    \r\n \t
                                  1. Yes, because [latex]P(C|D) = P(C)[\/latex].<\/li>\r\n \t
                                  2. No, because [latex]P(C \\text{ AND } D)[\/latex] is not equal to zero.<\/li>\r\n \t
                                  3. [latex]\\displaystyle{P}{({D}|{C})}=\\frac{{{P}{({C} \\text{ AND }{D})}}}{{{P}{({C})}}}=\\frac{{0.225}}{{0.75}}={0.3}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n
                                    \r\n

                                    try it<\/h3>\r\nA student goes to the library. Let events\u00a0[latex]B[\/latex] = the student checks out a book and [latex]D[\/latex] = the student checks out a DVD. Suppose that [latex]P(B) = 0.40[\/latex], [latex]P(D) = 0.30[\/latex] and [latex]P(B \\text{ AND } D) = 0.20[\/latex].\r\n
                                      \r\n \t
                                    1. Find [latex]P(B|D)[\/latex].<\/li>\r\n \t
                                    2. Find [latex]P(D|B)[\/latex].<\/li>\r\n \t
                                    3. Are [latex]B[\/latex] and [latex]D[\/latex] independent?<\/li>\r\n \t
                                    4. Are [latex]B[\/latex] and [latex]D[\/latex] mutually exclusive?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124086\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124086\"]\r\n
                                        \r\n \t
                                      1. [latex]P(B|D) = 0.6667[\/latex]<\/li>\r\n \t
                                      2. [latex]P(D|B) = 0.5[\/latex]<\/li>\r\n \t
                                      3. No<\/li>\r\n \t
                                      4. No<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
                                        \r\n

                                        Recall: Simplify a Fraction<\/h3>\r\nIt is a common convention in mathematics to present fractions in lowest terms. We call this practice simplifying or reducing the fraction, and it can be accomplished by canceling (dividing) the common factors in a fraction\u2019s numerator and denominator. We can do this because a fraction represents division, and for any number [latex]a, \\frac{a}{a}=1[\/latex].\r\n\r\n<\/div>\r\n
                                        \r\n

                                        Example<\/h3>\r\nIn a box there are three red cards and five blue cards. The red cards are marked with the numbers [latex]1, 2[\/latex], and [latex]3[\/latex], and the blue cards are marked with the numbers [latex]1, 2, 3, 4[\/latex], and [latex]5[\/latex]. The cards are well-shuffled. You reach into the box (you cannot see into it) and draw one card.\r\n\r\nLet\u00a0[latex]R[\/latex] = red card is drawn, [latex]B[\/latex] = blue card is drawn, [latex]E[\/latex] = even-numbered card is drawn.\r\n\r\nThe sample space\u00a0[latex]S = R1, R2, R3, B1, B2, B3, B4, B5[\/latex]. [latex]S[\/latex] has eight outcomes.\r\n
                                          \r\n \t
                                        • [latex]P(R) = \\displaystyle\\frac{{3}}{{8}}[\/latex].[latex]P(B) = \\displaystyle\\frac{{5}}{{8}}[\/latex]. [latex]P(R \\text{ AND } B) = 0[\/latex]. (You cannot draw one card that is both red and blue.)<\/li>\r\n \t
                                        • [latex]P(E) = \\displaystyle\\frac{{3}}{{8}}[\/latex]. (There are three even-numbered cards, [latex]R2, B2, \\text{ and } B4[\/latex].)<\/li>\r\n \t
                                        • [latex]P(E|B) = \\displaystyle\\frac{{2}}{{5}}[\/latex]. (There are five blue cards: [latex]B1, B2, B3, B4, \\text{ and } B5[\/latex]. Out of the blue cards, there are two even cards; [latex]B2 \\text{ and } B4[\/latex].)<\/li>\r\n \t
                                        • [latex]P(B|E) = \\displaystyle\\frac{{2}}{{3}}[\/latex]. (There are three even-numbered cards: [latex]R2, B2, \\text{ and } B4[\/latex]. Out of the even-numbered cards, two are blue; [latex]B2 \\text{ and }B4[\/latex].)<\/li>\r\n \t
                                        • The events\u00a0[latex]R[\/latex] and\u00a0[latex]B[\/latex] are mutually exclusive because [latex]P(R \\text{ AND } B) = 0[\/latex].<\/li>\r\n \t
                                        • Let [latex]G[\/latex] = card with a number greater than [latex]3[\/latex]. [latex]G[\/latex] = {[latex]B4, B5[\/latex]}. [latex]P(G) = \\displaystyle\\frac{{2}}{{8}}[\/latex]. Let H = blue card numbered between one and four, inclusive. [latex]H[\/latex] = {[latex]B1, B2, B3, B4[\/latex]}. [latex]P(G|H) = \\frac{1}{4}[\/latex]. (The only card in [latex]H[\/latex] that has a number greater than three is [latex]B4[\/latex].) Since [latex]\\frac{2}{8} = \\frac{1}{4}[\/latex], [latex]P(G) = P(G|H)[\/latex], which means that [latex]G[\/latex] and [latex]H[\/latex] are independent.<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n
                                          \r\n

                                          try it<\/h3>\r\nIn a basketball arena,\r\n
                                            \r\n \t
                                          • [latex]70[\/latex]% of the fans are rooting for the home team.<\/li>\r\n \t
                                          • [latex]25[\/latex]% of the fans are wearing blue.<\/li>\r\n \t
                                          • [latex]20[\/latex]% of the fans are wearing blue and are rooting for the away team.<\/li>\r\n \t
                                          • Of the fans rooting for the away team, [latex]67[\/latex]% are wearing blue.<\/li>\r\n<\/ul>\r\nLet\u00a0[latex]A[\/latex] be the event that a fan is rooting for the away team.\r\n\r\nLet\u00a0[latex]B[\/latex] be the event that a fan is wearing blue. Are the events of rooting for the away team and wearing blue independent? Are they mutually exclusive?\r\n[reveal-answer q=\"124087\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124087\"]\r\n\r\n[latex]P(B|A) = 0.67[\/latex]\r\n\r\n[latex]P(B) = 0.25[\/latex]\r\n\r\nSo\u00a0[latex]P(B)[\/latex] does not equal [latex]P(B|A)[\/latex] which means that [latex]B[\/latex] and [latex]A[\/latex] are not independent (wearing blue and rooting for the away team are not independent). They are also not mutually exclusive, because [latex]P(B \\text{ AND } A) = 0.20[\/latex], not [latex]0[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n
                                            \r\n

                                            Example<\/h3>\r\nIn a particular college class, [latex]60[\/latex]% of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, [latex]75[\/latex]% have long hair. Let\u00a0[latex]F[\/latex] be the event that a student is female. Let [latex]L[\/latex] be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?\r\n
                                              \r\n \t
                                            • The following probabilities are given in this example:<\/li>\r\n \t
                                            • [latex]P(F) = 0.60[\/latex]; [latex]P(L) = 0.50[\/latex]<\/li>\r\n \t
                                            • [latex]P(F \\text{ AND } L) = 0.45[\/latex]<\/li>\r\n \t
                                            • [latex]P(L|F) = 0.75[\/latex]<\/li>\r\n<\/ul>\r\nNote:\u00a0<\/strong>The choice you make depends on the information you have.<\/em> You could use the first or last condition on the list for this example. You do not know\u00a0[latex]P(F|L)[\/latex] yet, so you cannot use the second condition.\r\n[reveal-answer q=\"124088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124088\"]\r\n\r\nSolution 1:\r\n\r\nCheck whether\u00a0[latex]P(F \\text{ AND } L) = P(F)P(L)[\/latex]. We are given that [latex]P(F \\text{ AND } L) = 0.45[\/latex], but [latex]P(F)P(L) = (0.60)(0.50) = 0.30[\/latex]. The events of being female and having long hair are not independent because [latex]P(F \\text{ AND } L)[\/latex] does not equal [latex]P(F)P(L)[\/latex].\r\n\r\nSolution 2:\r\n\r\nCheck whether\u00a0[latex]P(L|F)[\/latex] equals [latex]P(L)[\/latex]. We are given that [latex]P(L|F) = 0.75[\/latex], but [latex]P(L) = 0.50[\/latex]; they are not equal. The events of being female and having long hair are not independent.\r\n\r\nInterpretation of Results:\r\nThe events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
                                              \r\n

                                              try it<\/h3>\r\nMark is deciding which route to take to work. His choices are\u00a0[latex]I[\/latex] = the Interstate and [latex]F[\/latex] = Fifth Street.\r\n
                                                \r\n \t
                                              • [latex]P(I) = 0.44[\/latex] and [latex]P(F) = 0.55[\/latex]<\/li>\r\n \t
                                              • [latex]P(I \\text{ AND } F) = 0[\/latex] because Mark will take only one route to work.<\/li>\r\n<\/ul>\r\nWhat is the probability of\u00a0[latex]P(I \\text{ OR } F)[\/latex]?\r\n[reveal-answer q=\"124089\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124089\"]\r\nBecause\u00a0[latex]P(I \\text{ AND } F) = 0[\/latex],\u00a0[latex]P(I \\text{ OR } F) = P(I) + P(F) - P(I AND F) = 0.44 + 0.56 - 0 = 1[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
                                                \r\n

                                                Recall: Fraction Multiplication<\/h3>\r\nIf [latex]a,b,c[\/latex], and [latex]d[\/latex] are numbers where [latex]b\\ne 0[\/latex] and [latex]d\\ne 0[\/latex] , then [latex]\\Large\\frac{a}{b}\\cdot \\frac{c}{d}\\normalsize=\\Large\\frac{ac}{bd}[\/latex].\r\n\r\n<\/div>\r\n
                                                \r\n

                                                Example<\/h3>\r\n
                                                  \r\n \t
                                                1. Toss one fair coin (the coin has two sides, [latex]H[\/latex] and [latex]T[\/latex]). The outcomes are ________. Count the outcomes. There are ____ outcomes.<\/li>\r\n \t
                                                2. Toss one fair, six-sided die (the die has [latex]1, 2, 3, 4, 5[\/latex], or [latex]6[\/latex] dots on a side). The outcomes are ________________. Count the outcomes. There are ___ outcomes.<\/li>\r\n \t
                                                3. Multiply the two numbers of outcomes. The answer is _______.<\/li>\r\n \t
                                                4. If you flip one fair coin and follow it with the toss of one fair, six-sided die, the answer in three is the number of outcomes (size of the sample space). What are the outcomes? (Hint: Two of the outcomes are [latex]H1[\/latex] and [latex]T6[\/latex].)<\/li>\r\n \t
                                                5. Event [latex]A[\/latex] = heads ([latex]H[\/latex]) on the coin followed by an even number ([latex]2, 4, 6[\/latex]) on the die.\r\n[latex]A[\/latex] = {_________________}. Find [latex]P(A)[\/latex].<\/li>\r\n \t
                                                6. Event [latex]B[\/latex] = heads on the coin followed by a three on the die. [latex]B[\/latex] = {________}. Find [latex]P(B)[\/latex].<\/li>\r\n \t
                                                7. Are [latex]A[\/latex] and [latex]B[\/latex] mutually exclusive? (Hint: What is [latex]P(A \\text{ AND } B)[\/latex]? If [latex]P(A \\text{ AND } B) = 0[\/latex], then [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive.)<\/li>\r\n \t
                                                8. Are [latex]A[\/latex] and [latex]B[\/latex] independent? (Hint: Is [latex]P(A \\text{ AND } B) = P(A)P(B)[\/latex]? If [latex]P(A \\text{ AND } B) = P(A)P(B)[\/latex], then [latex]A[\/latex] and [latex]B[\/latex] are independent. If not, then they are dependent).<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124090\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124090\"]\r\nSolution:\r\n
                                                    \r\n \t
                                                  1. [latex]H[\/latex] and [latex]T[\/latex]; 2<\/li>\r\n \t
                                                  2. [latex]1, 2, 3, 4, 5, 6; 6[\/latex]<\/li>\r\n \t
                                                  3. [latex]2(6) = 12[\/latex]<\/li>\r\n \t
                                                  4. [latex]T1, T2, T3, T4, T5, T6, H1, H2, H3, H4, H5, H6[\/latex]<\/li>\r\n \t
                                                  5. [latex]A[\/latex] = {[latex]H2, H4, H6[\/latex]}; [latex]P(A) = \\displaystyle\\frac{{3}}{{12}}[\/latex]<\/li>\r\n \t
                                                  6. [latex]B[\/latex] = {[latex]H3[\/latex]}; [latex]P(B) = \\displaystyle\\frac{{1}}{{12}}[\/latex]<\/li>\r\n \t
                                                  7. Yes, because [latex]P(A \\text{ AND } B) = 0[\/latex]<\/li>\r\n \t
                                                  8. [latex]P(A \\text{ AND } B) = 0[\/latex].[latex]P(A)P(B) = \\displaystyle{(\\frac{{3}}{{12}})}{(\\frac{{1}}{{12}})}[\/latex]. [latex]P(A \\text{ AND } B)[\/latex] does not equal [latex]P(A)P(B)[\/latex], so [latex]A[\/latex] and [latex]B[\/latex] are dependent.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n
                                                    \r\n

                                                    try it<\/h3>\r\nA box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let [latex]T[\/latex] be the event of getting the white ball twice, [latex]F[\/latex] the event of picking the white ball first, [latex]S[\/latex] the event of picking the white ball in the second drawing.\r\n
                                                      \r\n \t
                                                    1. Compute [latex]P(T)[\/latex].<\/li>\r\n \t
                                                    2. Compute [latex]P(T|F)[\/latex].<\/li>\r\n \t
                                                    3. Are [latex]T[\/latex] and [latex]F[\/latex] independent?.<\/li>\r\n \t
                                                    4. Are [latex]F[\/latex] and [latex]S[\/latex] mutually exclusive?<\/li>\r\n \t
                                                    5. Are [latex]F[\/latex] and [latex]S[\/latex] independent?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"124091\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124091\"]\r\n
                                                        \r\n \t
                                                      1. [latex]P(T) = \\displaystyle\\frac{{1}}{{4}}[\/latex]<\/li>\r\n \t
                                                      2. [latex]P(T|F) = \\displaystyle\\frac{{1}}{{2}}[\/latex]<\/li>\r\n \t
                                                      3. No<\/li>\r\n \t
                                                      4. No<\/li>\r\n \t
                                                      5. Yes<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"
                                                        \n

                                                        Learning Outcomes<\/h3>\n
                                                        \n
                                                          \n
                                                        • Determine if two events are mutually exclusive<\/li>\n
                                                        • Calculate probabilities for events that are mutually exclusive and events that are not mutually exclusive<\/li>\n
                                                        • Calculate probabilities to determine if two events are independent<\/li>\n<\/ul>\n<\/section>\n<\/div>\n

                                                          Mutually Exclusive Events<\/h2>\n

                                                          [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive<\/strong> events if they cannot occur at the same time. This means that [latex]A[\/latex] and [latex]B[\/latex] do not share any outcomes and [latex]P(A \\text{ AND } B) = 0[\/latex].<\/p>\n

                                                          For example, suppose the sample space\u00a0[latex]S[\/latex] = {[latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[\/latex]}. Let [latex]A[\/latex] = {[latex]1, 2, 3, 4, 5[\/latex]}, [latex]B[\/latex] = {[latex]4, 5, 6, 7, 8[\/latex]}, and [latex]C[\/latex] = {[latex]7, 9[\/latex]}. [latex]A \\text{ AND } B[\/latex] = {[latex]4, 5[\/latex]}. [latex]\\displaystyle{P}{({A} \\text{ AND } {B})}=\\frac{{2}}{{10}}[\/latex]and is not equal to zero. Therefore, [latex]A[\/latex] and [latex]B[\/latex] are not mutually exclusive. [latex]A[\/latex] and [latex]C[\/latex] do not have any numbers in common so [latex]P(A \\text{ AND } C) = 0[\/latex]. Therefore, [latex]A[\/latex] and [latex]C[\/latex] are mutually exclusive.<\/p>\n

                                                          If it is not known whether\u00a0[latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, assume they are not until you can show otherwise<\/strong>. The following examples illustrate these definitions and terms.<\/p>\n

                                                          <\/h3>\n
                                                          \n

                                                          Example<\/h3>\n

                                                          Flip two fair coins. (This is an experiment.)<\/p>\n

                                                          The sample space is {[latex]HH[\/latex], [latex]HT[\/latex], [latex]TH[\/latex], [latex]TT[\/latex]} where [latex]T[\/latex] = tails and [latex]H[\/latex] = heads. The outcomes are [latex]HH[\/latex], [latex]HT[\/latex], [latex]TH[\/latex], and [latex]TT[\/latex]. The outcomes HT and TH are different. The [latex]HT[\/latex] means that the first coin showed heads and the second coin showed tails. The [latex]TH[\/latex] means that the first coin showed tails and the second coin showed heads.<\/p>\n

                                                            \n
                                                          • Let [latex]A[\/latex] = the event of getting at most one tail<\/strong>. (At most one tail means zero or one tail.) Then [latex]A[\/latex] can be written as {[latex]HH[\/latex], [latex]HT[\/latex], [latex]TH[\/latex]}. The outcome [latex]HH[\/latex] shows zero tails. [latex]HT[\/latex] and [latex]TH[\/latex] each show one tail.<\/li>\n
                                                          • Let [latex]B[\/latex] = the event of getting all tails. [latex]B[\/latex] can be written as {[latex]TT[\/latex]}. [latex]B[\/latex] is the complement<\/strong> of [latex]A[\/latex], so [latex]B = A'[\/latex]. Also, [latex]P(A) + P(B)[\/latex] = [latex]P(A) + P(A') = 1[\/latex].<\/li>\n
                                                          • The probabilities for [latex]A[\/latex] and for [latex]B[\/latex] are [latex]P(A) = \\displaystyle\\frac{{3}}{{4}}[\/latex] and [latex]P(B) = \\displaystyle\\frac{{1}}{{4}}[\/latex].<\/li>\n
                                                          • Let [latex]C[\/latex] = the event of getting all heads. [latex]C[\/latex] = {[latex]HH[\/latex]}. Since [latex]B = {TT}[\/latex], [latex]P(B \\text{ AND } C) = 0[\/latex]. [latex]B[\/latex] and [latex]C[\/latex] are mutually exclusive. [latex]{(}B[\/latex] and [latex]C[\/latex] have no members in common because you cannot have all tails and all heads at the same time.)<\/li>\n
                                                          • Let [latex]D[\/latex] = event of getting more than one<\/strong> tail. [latex]D = {TT}[\/latex]. [latex]P(D) = \\displaystyle\\frac{{1}}{{4}}[\/latex]<\/li>\n
                                                          • Let [latex]E[\/latex] = event of getting a head on the first roll. (This implies you can get either a head or tail on the second roll.) [latex]E = {HT,HH}[\/latex]. [latex]P(E) = \\displaystyle\\frac{{2}}{{4}}[\/latex]<\/li>\n
                                                          • Find the probability of getting at least one<\/strong> (one or two) tail in two flips. Let [latex]F[\/latex] = event of getting at least one tail in two flips. [latex]F = {HT, TH, TT}[\/latex]. [latex]P(F) = \\displaystyle\\frac{{3}}{{4}}[\/latex]<\/li>\n<\/ul>\n<\/div>\n

                                                             <\/p>\n

                                                            <\/h3>\n
                                                            \n

                                                            try it<\/h3>\n

                                                            Draw two cards from a standard [latex]52[\/latex]-card deck with replacement. Find the probability of getting at least one black card.<\/p>\n

                                                            Show Solution<\/span><\/p>\n
                                                            \n

                                                            The sample space of drawing two cards with replacement from a standard 52-card deck with respect to color is {[latex]BB[\/latex], [latex]BR[\/latex], [latex]RB[\/latex], [latex]RR[\/latex]}.<\/p>\n

                                                            Event\u00a0[latex]A[\/latex] = Getting at least one black card = {[latex]BB[\/latex], [latex]BR[\/latex], [latex]RB[\/latex]}<\/p>\n

                                                            [latex]\\displaystyle{P}{({A})}=\\frac{{3}}{{4}}={0.75}[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n

                                                             <\/p>\n

                                                            <\/h3>\n
                                                            \n

                                                            Example<\/h3>\n

                                                            Flip two fair coins. Find the probabilities of the events.<\/p>\n

                                                              \n
                                                            1. Let [latex]F[\/latex] = the event of getting at most one tail (zero or one tail).<\/li>\n
                                                            2. Let [latex]G[\/latex] = the event of getting two faces that are the same.<\/li>\n
                                                            3. Let [latex]H[\/latex] = the event of getting a head on the first flip followed by a head or tail on the second flip.<\/li>\n
                                                            4. Are [latex]F[\/latex] and [latex]G[\/latex] mutually exclusive?<\/li>\n
                                                            5. Let [latex]J[\/latex] = the event of getting all tails. Are [latex]J[\/latex] and [latex]H[\/latex] mutually exclusive?<\/li>\n<\/ol>\n
                                                              Show Solution<\/span><\/p>\n
                                                              \n

                                                              Solution:<\/p>\n

                                                              Look at the sample space in Example 3.<\/p>\n

                                                                \n
                                                              1. Zero ([latex]0[\/latex]) or one ([latex]1[\/latex]) tails occur when the outcomes [latex]HH[\/latex], [latex]TH[\/latex], [latex]HT[\/latex] show up. [latex]P(F) = \\displaystyle\\frac{{3}}{{4}}[\/latex]<\/li>\n
                                                              2. Two faces are the same if [latex]HH[\/latex] or [latex]TT[\/latex] show up. [latex]P(G) = \\displaystyle\\frac{{2}}{{4}}[\/latex]<\/li>\n
                                                              3. A head on the first flip followed by a head or tail on the second flip occurs when [latex]HH[\/latex] or [latex]HT[\/latex] show up. [latex]P(H) = \\displaystyle\\frac{{2}}{{4}}[\/latex]<\/li>\n
                                                              4. [latex]F[\/latex] and [latex]G[\/latex] share [latex]HH[\/latex] so [latex]P(F \\text{ AND } G)[\/latex] is not equal to zero ([latex]0[\/latex]). [latex]F[\/latex] and [latex]G[\/latex] are not mutually exclusive.<\/li>\n
                                                              5. Getting all tails occurs when tails shows up on both coins ([latex]TT[\/latex]). [latex]H[\/latex]‘s outcomes are [latex]HH[\/latex] and [latex]HT[\/latex]. [latex]J[\/latex] and [latex]H[\/latex] have nothing in common so [latex]P(J \\text{ AND } H) = 0[\/latex]. [latex]J[\/latex] and [latex]H[\/latex] are mutually exclusive.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

                                                                 <\/p>\n

                                                                \n

                                                                This video provides two more examples of finding the probability of events that are mutually exclusive.<\/p>\n