BTGmoderatorDC wrote:Pam and Robin each roll a pair of fair, six-sided dice. What is the probability that Pam and Robin will both roll the same set of two numbers?
A. 1/216
B. 1/36
C. 5/108
D. 11/216
E. 1/18
OA D
Source: Veritas Prep
Pam can obtain either of: (1) a pair of the same two numbers, or (2) a pair of two different numbers. We have to consider each case separately because the probability of Robin matching is different for each case. Here's why: If Pam rolls a (2,2), for example, then Robin has only 1 chance in 36 of matching. But if Pam rolls a (6,2), for example, then Robin can match by rolling either a (6,2) or a (2,6), so the probability is 2/36, or 1/18.
If Pam rolls a pair of same two numbers, she has a probability of 1 x 1/6 = 1/6 of rolling them. Then Robin has a probability of 1/6 x 1/6 = 1/36 to match them. So the probability they will roll the same set of two numbers when Pam rolls a pair of same numbers is 1/6 x 1/36 = 1/216.
If Pam rolls a pair of different numbers, she has a probability of 1 x 5/6 = 5/6 of rolling them. Then Robin has a 1/6 x 1/6 = 1/36 chance for her first number to match Pam's first number and her second number to match Pam's second number, and another 1/6 x 1/6 = 1/36 chance for her first number to match Pam's second number and her second number to match Pam's first number. Therefore, Robin has a 2/36 chance to match Pam's numbers. So the probability they will roll the same set of two numbers when Pam rolls a pair of different numbers is 5/6 x 2/36 = 10/216.
Therefore, the probability that they will roll the same set of two numbers 1/216 + 10/216 = 11/216.
Alternate Solution:
Suppose Pam rolls the first die. There are two cases: a) Robin's first die will match Pam's first die (a probability of 1/6) or b) Robin's first die will not match Pam's first die (a probability of 5/6).
In the first case, the only way they will get the same set of numbers is if Robin's second roll matches Pam's second roll; which has a probability of 1/6. Since the probability of this case was 1/6, the probability of obtaining a matched pair from this scenario is 1/6 x 1/6 = 1/36.
In the second case, to obtain a matched pair, Pam's second roll must match Robin's first roll (a probability of 1/6) and Robin's second roll must match Pam's first roll (a probability of 1/6). Since the probability of this case was 5/6, the probability of obtaining a matched pair from this scenario is 1/6 x 1/6 x 5/6 = 5/216.
We note that the two scenarios are mutually exclusive events. Therefore, the total probability of obtaining a matched pair is simply the sum of the individual probabilities of obtaining a matched pair from each case, which is 1/36 + 5/216 = 6/216 + 5/216 = 11/216.
Answer: D
