{"id":5057,"date":"2022-08-17T20:03:53","date_gmt":"2022-08-17T20:03:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5057"},"modified":"2022-08-17T20:48:26","modified_gmt":"2022-08-17T20:48:26","slug":"8c-coreq","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/8c-coreq\/","title":{"raw":"8C Coreq","rendered":"8C Coreq"},"content":{"raw":"In the next preview assignment and in the next class, you will need to compute probabilities that involve exponents, and you will need to interpret statements of inequality and use those to find probabilities.\r\nHow Many 1\u2019s?\r\nWhen events [latex] A [\/latex] and [latex] B [\/latex] are independent, then [latex] P(A\\mbox{ and }B) = P(A)\u2219P(B) [\/latex].\r\nWhen using this rule, there are situations in which the probabilities of some events are the same, and that\u2019s when exponents come into play.\r\nExponent Review: [latex] p^{x} [\/latex] means that [latex] p [\/latex] is multiplied by itself [latex] x [\/latex] times. For example, [latex] (0.5)^{2} = (0.5) \\bullet (0.5) = 0.25, \\mbox{ and } (0.5)^{3} = (0.5) \\bullet (0.5) \\bullet (0.5) = 0.125 [\/latex].\r\nRecall that multiplication is commutative, which means that the order we multiply numbers in does not matter when multiplication is the only operation present. For example:\r\n[latex] (0.25) \\bullet (0.5) \\bullet (0.25) = (0.25) \\bullet (0.25) \\bullet (0.5) = (0.25)^{2}(0.5) [\/latex]\r\n\r\nExample: Suppose we are rolling a fair, 6-sided die 4 times. The probability that the first roll is not a 6, the second roll is a 6, and the final 2 rolls are also not 6 is:\r\n[latex] P(not~6~AND~6~AND~not~6~AND~not~6) = P(not~6) * P(6) * P(not~6) * P(not~6) = P(6) \\bullet (P(not~6))^{3} = (\\frac{1}{6}) \\bullet (\\frac{5}{6})^{3} \\approx 0.0965 [\/latex]\r\n
\r\n

Question 1<\/h3>\r\nSuppose that we have a spinner with 4 sections, each of equal size, and the sections are labeled 1, 2, 3, and 4. Then the probability of the arrow landing in any one section on a spin is [latex] \\frac{1}{4} =0.25 [\/latex]. Notice also that the outcomes of different spins are independent, so the multiplication rule for independent events applies here.\r\n\r\n\"\"\r\n
    \r\n \t
  1. If we spin the spinner twice, what is the probability that the arrow lands on the 1 section both times?<\/li>\r\n \t
  2. What is the probability that the arrow does not land on the 1 section in a single spin of the spinner?<\/li>\r\n \t
  3. If we spin the spinner 3 times, what is the probability that on the first 2 spins, the arrow lands on the 1 section, but on the third spin, the arrow does not land on the 1 section?<\/li>\r\n \t
  4. If we spin the spinner 3 times, what is the probability that on the first spin, the arrow does not land on the 1 section, but on the next 2 spins, the arrow does land on the 1 section?<\/li>\r\n \t
  5. What do you notice about the resulting probabilities from Parts C and D?<\/li>\r\n \t
  6. If we spin the spinner 10 times, what is the probability that on the second spin, the arrow lands on 1, but on all of the other spins, the arrow does not land on 1? Hint: Since there are 10 spins of the spinner, first think about writing your probability as an AND statement that indicates what happens with each spin. The probabilities of each spin would then need to be multiplied together (why?). How many spins land on 1 and how many spins do not land on 1?\r\n
      \r\n \t
    1. [latex] (0.25)^{9}(0.75) [\/latex]<\/li>\r\n \t
    2. [latex] (0.25)^{8}(0.75)^{2} [\/latex]<\/li>\r\n \t
    3. [latex] (0.25)(0.75)^{9}[\/latex]<\/li>\r\n \t
    4. [latex] (0.25)^{2}(0.75)^{8}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the next preview assignment, you will be considering probability questions where each trial of an experiment has exactly two outcomes (which we call \u201csuccess\u201d and \u201cfailure\u201d), and we want to count how many of the trials result in successes.\r\n\r\nIn the context of our spinner question, we might want to count how many times the spinner lands on the 1 section if we spin the spinner 10 times. We will also consider questions like, \u201cWhat is the probability that in at least 6 out of 10 spin attempts, the arrow lands on the 1 section?\u201d For these types of questions, we will need inequality expressions.\r\n\r\nInequality Review - There are four inequality symbols:\r\n
        \r\n \t
      • [latex]<[\/latex] (less than): [latex] a<b [\/latex] means that \u201ca is less than b.\u201d For example, we write [latex] 2 < 5 [\/latex] because 2 is less than 5.<\/li>\r\n \t
      • [latex]\\leq[\/latex] (less than or equal to): [latex] a \\leq b [\/latex] means that \u201ca is less than or equal to b.\u201d For example, we can write [latex]2 \\leq 5 [\/latex] and [latex]5 \\leq 5 [\/latex] because the symbol indicates that there are two possibilities: the number on the left is either less than 5 or the number on the left equals 5.<\/li>\r\n \t
      • [latex]>[\/latex] (greater than): [latex] a>b [\/latex] means that \u201ca is greater than b.\u201d For example, we write [latex]5 > 2 [\/latex] because 5 is greater than 2.<\/li>\r\n \t
      • [latex]\\geq[\/latex] (greater than or equal to): [latex]a \\geq b [\/latex] means that \u201ca is greater than or equal to b.\u201d For example, we can write [latex]5 \\geq 2 [\/latex] and [latex]5 \\geq 5 [\/latex] because the symbol indicates that there are two possibilities: the number on the left is either greater than 5 or the number on the left equals 5.<\/li>\r\n<\/ul>\r\nOne way to help remember how these symbols work is that the pointy end (the smaller side) of the symbol always points to the smaller number, while the open side (the bigger side) of the symbol always opens to the bigger number. (A fun way to remember it is to think of the symbol as a mouth that wants to eat the biggest number!)\r\n\r\nAn important part of answering probability questions is translating the question into a mathematical expression so that you know what probability you\u2019re trying to find.\r\n\r\nExample: As above, let\u2019s imagine we are spinning a spinner with 4 equally-sized sections 10 times. Let the random variable [latex] X [\/latex]be the number of times that we land on the 1 in our 10 spin attempts. Then we can describe the probability that we land on 1 on more than 5 spins as [latex]P(X> 5) [\/latex] since we want the number of spins where the arrow lands on 1 to be more than 5.\r\n
        \r\n

        Question 2<\/h3>\r\nSelect the mathematical expression that corresponds to each of the following probability descriptions.\r\n
          \r\n \t
        1. The probability that the spinner lands on 1 on fewer than 3 spins\r\n
            \r\n \t
          1. [latex] P(X<3) [\/latex]<\/li>\r\n \t
          2. [latex] P(X \\leq 3) [\/latex]<\/li>\r\n \t
          3. [latex] P(X > 3) [\/latex]<\/li>\r\n \t
          4. [latex] P(X \\geq 3) [\/latex]<\/li>\r\n \t
          5. [latex] P(X = 3) [\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
          6. The probability that the spinner lands on 1 on exactly 3 spins\r\n
              \r\n \t
            1. [latex] P(X<3) [\/latex]<\/li>\r\n \t
            2. [latex] P(X \\leq 3) [\/latex]<\/li>\r\n \t
            3. [latex] P(X > 3) [\/latex]<\/li>\r\n \t
            4. [latex] P(X \\geq 3) [\/latex]<\/li>\r\n \t
            5. [latex] P(X = 3) [\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
            6. The probability that the spinner lands on 1 on 3 or more spins\r\n
                \r\n \t
              1. [latex] P(X<3) [\/latex]<\/li>\r\n \t
              2. [latex] P(X \\leq 3) [\/latex]<\/li>\r\n \t
              3. [latex] P(X > 3) [\/latex]<\/li>\r\n \t
              4. [latex] P(X \\geq 3) [\/latex]<\/li>\r\n \t
              5. [latex] P(X = 3) [\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\nSometimes, a description of an inequality can be a little trickier to interpret.\r\n
                  \r\n \t
                • \u201cAt least x\u201d means x or more. For example, \u201cat least 3\u201d means 3 or more.<\/li>\r\n \t
                • \u201cUp to x\u201d means less than or equal to x. For example, \u201cup to 2\u201d means 2 or less.<\/li>\r\n \t
                • \u201cAt most x\u201d means less than or equal to x. For example, \u201cat most 2\u201d means 2 or less.<\/li>\r\n \t
                • \u201cNo more than x\u201d means less than or equal to x. For example, \u201cno more than 2\u201d means 2 or less.<\/li>\r\n<\/ul>\r\n
                  \r\n

                  Question 3<\/h3>\r\nFor each description, say how many possible spins out of 10 are being discussed. For example, \u201cat most 3 spins land on 1\u201d would mean that \u201c0, 1, 2, or 3 spins land on 1.\u201d\r\n
                    \r\n \t
                  1. Up to 3 spins land on 1<\/li>\r\n \t
                  2. At least 7 spins land on 1<\/li>\r\n \t
                  3. No more than 3 spins land on 1<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                    \r\n

                    Question 4<\/h3>\r\nFor each probability, select the corresponding mathematical expression where, as previously, X is the number of times the spinner lands on 1 out of 10 spins.\r\n
                      \r\n \t
                    1. The probability that the spinner lands on 1 on at least 5 spins\r\n
                        \r\n \t
                      1. [latex] P(X<5) [\/latex]<\/li>\r\n \t
                      2. [latex] P(X \\leq 5) [\/latex]<\/li>\r\n \t
                      3. [latex] P(X > 5) [\/latex]<\/li>\r\n \t
                      4. [latex] P(X \\geq 5) [\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                      5. The probability that the spinner lands on 1 on at most 5 spins\r\n
                          \r\n \t
                        1. [latex] P(X<5) [\/latex]<\/li>\r\n \t
                        2. [latex] P(X \\leq 5) [\/latex]<\/li>\r\n \t
                        3. [latex] P(X > 5) [\/latex]<\/li>\r\n \t
                        4. [latex] P(X \\geq 5) [\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                        5. The probability that the spinner lands on 1 on no more than 5 spins\r\n
                            \r\n \t
                          1. [latex] P(X<5) [\/latex]<\/li>\r\n \t
                          2. [latex] P(X \\leq 5) [\/latex]<\/li>\r\n \t
                          3. [latex] P(X > 5) [\/latex]<\/li>\r\n \t
                          4. [latex] P(X \\geq 5) [\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n ","rendered":"

                            In the next preview assignment and in the next class, you will need to compute probabilities that involve exponents, and you will need to interpret statements of inequality and use those to find probabilities.
                            \nHow Many 1\u2019s?
                            \nWhen events [latex]A[\/latex] and [latex]B[\/latex] are independent, then [latex]P(A\\mbox{ and }B) = P(A)\u2219P(B)[\/latex].
                            \nWhen using this rule, there are situations in which the probabilities of some events are the same, and that\u2019s when exponents come into play.
                            \nExponent Review: [latex]p^{x}[\/latex] means that [latex]p[\/latex] is multiplied by itself [latex]x[\/latex] times. For example, [latex](0.5)^{2} = (0.5) \\bullet (0.5) = 0.25, \\mbox{ and } (0.5)^{3} = (0.5) \\bullet (0.5) \\bullet (0.5) = 0.125[\/latex].
                            \nRecall that multiplication is commutative, which means that the order we multiply numbers in does not matter when multiplication is the only operation present. For example:
                            \n[latex](0.25) \\bullet (0.5) \\bullet (0.25) = (0.25) \\bullet (0.25) \\bullet (0.5) = (0.25)^{2}(0.5)[\/latex]<\/p>\n

                            Example: Suppose we are rolling a fair, 6-sided die 4 times. The probability that the first roll is not a 6, the second roll is a 6, and the final 2 rolls are also not 6 is:
                            \n[latex]P(not~6~AND~6~AND~not~6~AND~not~6) = P(not~6) * P(6) * P(not~6) * P(not~6) = P(6) \\bullet (P(not~6))^{3} = (\\frac{1}{6}) \\bullet (\\frac{5}{6})^{3} \\approx 0.0965[\/latex]<\/p>\n

                            \n

                            Question 1<\/h3>\n

                            Suppose that we have a spinner with 4 sections, each of equal size, and the sections are labeled 1, 2, 3, and 4. Then the probability of the arrow landing in any one section on a spin is [latex]\\frac{1}{4} =0.25[\/latex]. Notice also that the outcomes of different spins are independent, so the multiplication rule for independent events applies here.<\/p>\n

                            \"\"<\/p>\n

                              \n
                            1. If we spin the spinner twice, what is the probability that the arrow lands on the 1 section both times?<\/li>\n
                            2. What is the probability that the arrow does not land on the 1 section in a single spin of the spinner?<\/li>\n
                            3. If we spin the spinner 3 times, what is the probability that on the first 2 spins, the arrow lands on the 1 section, but on the third spin, the arrow does not land on the 1 section?<\/li>\n
                            4. If we spin the spinner 3 times, what is the probability that on the first spin, the arrow does not land on the 1 section, but on the next 2 spins, the arrow does land on the 1 section?<\/li>\n
                            5. What do you notice about the resulting probabilities from Parts C and D?<\/li>\n
                            6. If we spin the spinner 10 times, what is the probability that on the second spin, the arrow lands on 1, but on all of the other spins, the arrow does not land on 1? Hint: Since there are 10 spins of the spinner, first think about writing your probability as an AND statement that indicates what happens with each spin. The probabilities of each spin would then need to be multiplied together (why?). How many spins land on 1 and how many spins do not land on 1?\n
                                \n
                              1. [latex](0.25)^{9}(0.75)[\/latex]<\/li>\n
                              2. [latex](0.25)^{8}(0.75)^{2}[\/latex]<\/li>\n
                              3. [latex](0.25)(0.75)^{9}[\/latex]<\/li>\n
                              4. [latex](0.25)^{2}(0.75)^{8}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n

                                In the next preview assignment, you will be considering probability questions where each trial of an experiment has exactly two outcomes (which we call \u201csuccess\u201d and \u201cfailure\u201d), and we want to count how many of the trials result in successes.<\/p>\n

                                In the context of our spinner question, we might want to count how many times the spinner lands on the 1 section if we spin the spinner 10 times. We will also consider questions like, \u201cWhat is the probability that in at least 6 out of 10 spin attempts, the arrow lands on the 1 section?\u201d For these types of questions, we will need inequality expressions.<\/p>\n

                                Inequality Review – There are four inequality symbols:<\/p>\n

                                  \n
                                • [latex]<[\/latex] (less than): [latex]a\n
                                • [latex]\\leq[\/latex] (less than or equal to): [latex]a \\leq b[\/latex] means that \u201ca is less than or equal to b.\u201d For example, we can write [latex]2 \\leq 5[\/latex] and [latex]5 \\leq 5[\/latex] because the symbol indicates that there are two possibilities: the number on the left is either less than 5 or the number on the left equals 5.<\/li>\n
                                • [latex]>[\/latex] (greater than): [latex]a>b[\/latex] means that \u201ca is greater than b.\u201d For example, we write [latex]5 > 2[\/latex] because 5 is greater than 2.<\/li>\n
                                • [latex]\\geq[\/latex] (greater than or equal to): [latex]a \\geq b[\/latex] means that \u201ca is greater than or equal to b.\u201d For example, we can write [latex]5 \\geq 2[\/latex] and [latex]5 \\geq 5[\/latex] because the symbol indicates that there are two possibilities: the number on the left is either greater than 5 or the number on the left equals 5.<\/li>\n<\/ul>\n

                                  One way to help remember how these symbols work is that the pointy end (the smaller side) of the symbol always points to the smaller number, while the open side (the bigger side) of the symbol always opens to the bigger number. (A fun way to remember it is to think of the symbol as a mouth that wants to eat the biggest number!)<\/p>\n

                                  An important part of answering probability questions is translating the question into a mathematical expression so that you know what probability you\u2019re trying to find.<\/p>\n

                                  Example: As above, let\u2019s imagine we are spinning a spinner with 4 equally-sized sections 10 times. Let the random variable [latex]X[\/latex]be the number of times that we land on the 1 in our 10 spin attempts. Then we can describe the probability that we land on 1 on more than 5 spins as [latex]P(X> 5)[\/latex] since we want the number of spins where the arrow lands on 1 to be more than 5.<\/p>\n

                                  \n

                                  Question 2<\/h3>\n

                                  Select the mathematical expression that corresponds to each of the following probability descriptions.<\/p>\n

                                    \n
                                  1. The probability that the spinner lands on 1 on fewer than 3 spins\n
                                      \n
                                    1. [latex]P(X<3)[\/latex]<\/li>\n
                                    2. [latex]P(X \\leq 3)[\/latex]<\/li>\n
                                    3. [latex]P(X > 3)[\/latex]<\/li>\n
                                    4. [latex]P(X \\geq 3)[\/latex]<\/li>\n
                                    5. [latex]P(X = 3)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                    6. The probability that the spinner lands on 1 on exactly 3 spins\n
                                        \n
                                      1. [latex]P(X<3)[\/latex]<\/li>\n
                                      2. [latex]P(X \\leq 3)[\/latex]<\/li>\n
                                      3. [latex]P(X > 3)[\/latex]<\/li>\n
                                      4. [latex]P(X \\geq 3)[\/latex]<\/li>\n
                                      5. [latex]P(X = 3)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                      6. The probability that the spinner lands on 1 on 3 or more spins\n
                                          \n
                                        1. [latex]P(X<3)[\/latex]<\/li>\n
                                        2. [latex]P(X \\leq 3)[\/latex]<\/li>\n
                                        3. [latex]P(X > 3)[\/latex]<\/li>\n
                                        4. [latex]P(X \\geq 3)[\/latex]<\/li>\n
                                        5. [latex]P(X = 3)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n

                                          Sometimes, a description of an inequality can be a little trickier to interpret.<\/p>\n

                                            \n
                                          • \u201cAt least x\u201d means x or more. For example, \u201cat least 3\u201d means 3 or more.<\/li>\n
                                          • \u201cUp to x\u201d means less than or equal to x. For example, \u201cup to 2\u201d means 2 or less.<\/li>\n
                                          • \u201cAt most x\u201d means less than or equal to x. For example, \u201cat most 2\u201d means 2 or less.<\/li>\n
                                          • \u201cNo more than x\u201d means less than or equal to x. For example, \u201cno more than 2\u201d means 2 or less.<\/li>\n<\/ul>\n
                                            \n

                                            Question 3<\/h3>\n

                                            For each description, say how many possible spins out of 10 are being discussed. For example, \u201cat most 3 spins land on 1\u201d would mean that \u201c0, 1, 2, or 3 spins land on 1.\u201d<\/p>\n

                                              \n
                                            1. Up to 3 spins land on 1<\/li>\n
                                            2. At least 7 spins land on 1<\/li>\n
                                            3. No more than 3 spins land on 1<\/li>\n<\/ol>\n<\/div>\n
                                              \n

                                              Question 4<\/h3>\n

                                              For each probability, select the corresponding mathematical expression where, as previously, X is the number of times the spinner lands on 1 out of 10 spins.<\/p>\n

                                                \n
                                              1. The probability that the spinner lands on 1 on at least 5 spins\n
                                                  \n
                                                1. [latex]P(X<5)[\/latex]<\/li>\n
                                                2. [latex]P(X \\leq 5)[\/latex]<\/li>\n
                                                3. [latex]P(X > 5)[\/latex]<\/li>\n
                                                4. [latex]P(X \\geq 5)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                5. The probability that the spinner lands on 1 on at most 5 spins\n
                                                    \n
                                                  1. [latex]P(X<5)[\/latex]<\/li>\n
                                                  2. [latex]P(X \\leq 5)[\/latex]<\/li>\n
                                                  3. [latex]P(X > 5)[\/latex]<\/li>\n
                                                  4. [latex]P(X \\geq 5)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                  5. The probability that the spinner lands on 1 on no more than 5 spins\n
                                                      \n
                                                    1. [latex]P(X<5)[\/latex]<\/li>\n
                                                    2. [latex]P(X \\leq 5)[\/latex]<\/li>\n
                                                    3. [latex]P(X > 5)[\/latex]<\/li>\n
                                                    4. [latex]P(X \\geq 5)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n

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