Alice throws a die 12 times then gives it to Bob, Bob throws the die 6 times. They then honestly report that the die landed 3 times on 1. What is the probability that Alice will get 1 on exactly 2 times rolling on 12 times rolling?

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Simeon Hester · Accepted Answer

Step 1The probability that Alice will get 1 on exactly 2 times rolling on 12 times rolling.                                          (                                    12          2                                      )                              ×              5                  10                            6              12              =            66      ×              5                  10                            6              12              =  0.2961Step 2                    (                    12      2                      )              ×      5          10       is the number of possible outcomes of two ones if rolling a die 12 times

gravplasszo · Accepted Answer

Step 1We know that a 1 is obtained on exactly 3 of the 18 rolls of the die, which gives an experimental probability of obtaining a 1 of   p  =      3    18    =      1    6   which happens to also be the probability of obtaining a 1 when a fair die is rolled.Step 2We want to find the probability that a 1 is obtained on exactly two of the first twelve rolls. The geometric distribution is used to find the probability that it will take X Bernoulli trials to obtain the first success for an event with probability p, which is not what we need here. Instead, we want to use the Binomial distribution, which gives the probability that exactly k successes will occur in n trials of an experiment if each trial has probability p of success. The probability of exactly k successes in n trials is given by the formula   Pr  (  X  =  k  )  =                    (                    n      k                      )                  p    k    (  1  −  p      )          n      −      k       where                     (                    n      k                      )             is the number of ways exactly k successes can occur in n trials,       p    k   is the probability of k successes, and   (  1  −  p      )          n      −      k       is the probability of   n  −  k failures.Step 3In this problem,   n  =  12 and   k  =  2. Hence, the probability that exactly two of the twelve rolls that Alice makes result in the outcome 1 is   Pr  (  X  =  2  )  =                    (                    12      2                      )                        (              1        6            )        2              (              5        6            )              10

Alice throws a die 12 times then gives it to Bob, Bob throws the die 6 times. They then honestly report that the die landed 3 times on 1. What is the probability that Alice will get 1 on exactly 2 times rolling on 12 times rolling?

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