{"id":5031,"date":"2022-08-17T19:06:03","date_gmt":"2022-08-17T19:06:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5031"},"modified":"2022-08-17T19:16:07","modified_gmt":"2022-08-17T19:16:07","slug":"8b-preview","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/8b-preview\/","title":{"raw":"8B Preview","rendered":"8B Preview"},"content":{"raw":"Preparing for the next class\r\n\r\nIn the next in-class activity, you will need to make connections between frequency,\u00a0 relative frequency, and probability. You will also need to interpret discrete probability\u00a0 distributions to calculate probabilities, including those involving the phrases OR, at\u00a0 least, at most, less than, and greater than. Finally, you will need to identify whether a\u00a0 table or graph represents a probability distribution for a discrete random variable.\r\n
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Question 1<\/h3>\r\nConsider the chance experiment of flipping a fair coin 3 times. The 8 possible outcomes of this experiment are provided below (with \u201cH\u201d representing heads and\u00a0 \u201cT\u201d representing tails):\r\n\r\nHHH HHT HTH THH TTH THT HTT TTT\r\n
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  1. Suppose we are interested in describing the number of tails that are flipped. Using the random variable [latex] X [\/latex] to represent the number of tails that we find in\u00a0 each outcome, complete the following table.\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Experimental Outcome<\/td>\r\n[latex] X [\/latex], Number of Tails in\u00a0 3 Flips of a Coin<\/td>\r\n<\/tr>\r\n
    HHH<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
    HHT<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    HTH<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    THH<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    TTH<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    THT<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    HTT<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    TTT<\/td>\r\n3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nHint: The variable\u00a0[latex] X [\/latex] represents the number of tails we find in 3 coin flips. The outcome HHH has 0 tails, and TTT has 3 tails. Count the number of tails in each of\u00a0 the remaining outcomes of the experiment to complete the table.<\/li>\r\n \t
  2. \u00a0In the second column of the previous table, we can see that our random\u00a0 variable, [latex] X [\/latex], could be one of four values depending on the number of tails that\u00a0 are flipped (0, 1, 2, or 3).Out of the 8 possible experimental outcomes, how many times do we find 0\u00a0 tails? What about for the other values of the random variable (1, 2, and 3)?\u00a0 Complete the following table with the frequency and relative frequency\u00a0 values.Remember, relative frequency can be calculated by taking the frequency of\u00a0 the individual value and dividing it by the total frequency.\r\n
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    \u00a0[latex] X [\/latex], Number of Tails in 3\u00a0 Flips of a Coin<\/td>\r\nFrequency<\/td>\r\nRelative\r\n\r\nFrequency<\/td>\r\n<\/tr>\r\n
    0<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    1<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    2<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    3<\/td>\r\n1<\/td>\r\n0.125<\/td>\r\n<\/tr>\r\n
    Total:<\/td>\r\n8<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nHint: Go through each of the 8 possible outcomes provided in the table in Part A and\u00a0 count the number of the tails that are flipped. How many of the 8 outcomes have no\u00a0 tails? This is the frequency associated with 0 tails. How many of the 8 outcomes\u00a0 have exactly 1 tails? This is the frequency for 1. Repeat this for 2 tails and 3 tails,\u00a0 and then enter these values in the table.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the first column of our table, we see the set of possible values of our random variable, [latex] X [\/latex]. This variable\u00a0[latex] X [\/latex] is classified as a discrete random variable because it takes a fixed\u00a0 set of possible numerical values and it is not possible to get any value in between.\r\n\r\nThe probability distribution of a discrete random variable\u00a0[latex] X [\/latex] describes all possible\u00a0 values of the random variable, as well as the probability associated with each value. A\u00a0 discrete probability distribution is often presented in a table or graph.\r\n
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    Question 2<\/h3>\r\nDiscrete probability distributions can be represented with a table or bar graph. The following graph represents the probability distribution for the discrete variable\u00a0 number of tails in 3 flips of a coin. The graph includes the possible outcomes of the\u00a0 discrete variable and the probability associated with each outcome. We can see the\u00a0 probabilities are identical to the relative frequency values we calculated in the\u00a0 previous table.\r\n\r\n\"\"\r\n\r\nWe can use this graph to calculate probabilities associated with values of the\u00a0 random variable.\r\n
      \r\n \t
    1. According to the probability distribution table, the probability associated with\u00a0 the value \u201c2\u201d is 0.375. This can also be written as [latex]P(X = 2)[\/latex] = 0.375 or\u00a0 [latex] P [\/latex](2 tails) = 0.375. Which of the following is the correct interpretation of this\u00a0 mathematical statement?\r\n
        \r\n \t
      1. The probability of getting 3 tails in 3 coin flips is 0.375.<\/li>\r\n \t
      2. The probability of getting 2 tails in 3 coin flips is 0.375.<\/li>\r\n \t
      3. The probability of getting this result in 2 coin flips is 0.375.<\/li>\r\n \t
      4. The probability of getting 0, 1, 2, or 3 tails in 3 flips is 0.375.\u00a0 Hint: The number inside the parentheses describes the value of the random variable,\u00a0 described in the first column of our probability distribution table.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      5. According to the probability distribution table, what is the probability of\u00a0 flipping all tails? In other words, what is [latex]P(X = 3)[\/latex] or [latex] P [\/latex](3 tails)?\r\nHint: The number inside the parentheses describes the value of the random variable,\u00a0 described in the first column of our probability distribution table.<\/li>\r\n \t
      6. The values of the random variable, [latex] X [\/latex], are mutually exclusive. This means\u00a0 we can use what we learned in In-Class Activity 7.B to calculate probabilities\u00a0 involving the word \u201cOR:\"\r\n[latex] P(A OR B) = P(A) + P(B) [\/latex]\r\nThe probability of finding at least 2 tails in 3 coin flips, written[latex] P[\/latex](2 tails OR 3 tails), can be calculated by [latex] P[\/latex](2 tails) + [latex] P[\/latex](3 tails).Calculate this value.Hint: Identify the probability of getting 2 tails,[latex] P[\/latex](2 tails), and the probability of getting\u00a0 3 tails, [latex] P[\/latex](3 tails). [latex] P[\/latex](2 tails OR 3 tails) =[latex] P[\/latex](2 tails) + [latex] P[\/latex](3 tails)<\/li>\r\n \t
      7. \u00a0According to the probability distribution table, what is the probability of\u00a0 flipping less than 2 tails? That is, what is [latex] P(X < 2) [\/latex]?\r\nHint: Flipping less than 2 tails means we would flip either 0 or 1 tails. What is\u00a0 [latex] P [\/latex](0 tails OR 1 tails)?<\/li>\r\n<\/ol>\r\n<\/div>\r\nThere are a few important properties of discrete probability distributions that we\u00a0 should consider:\r\n