Final answer:
The probability that Sue picks two shoes of the same color and one is a left shoe and the other is a right shoe is 1/3. The total possible shoe combinations are calculated and compared to the number of combinations representing matching pairs of different colors to find this probability.
Step-by-step explanation:
The question asks us to calculate the probability that Sue picks two shoes of the same color, with one being a left shoe and the other a right shoe. Since shoes come in pairs, there is already an inherent left and right shoe for each pair. Therefore, we don't need to consider left and right in our probability calculation directly since pairing a left with a right is the only way to get a proper pair. However, the condition is implicit in the scenario that one of each must be chosen.
First, we calculate the total number of ways in which two shoes can be picked from all available shoes. Sue has 11 pairs, or 22 individual shoes. The first shoe can be any of the 22, and the second can be any of the remaining 21. Thus, the total number of ways to pick two shoes is 22 P 2 = 22 * 21.
Now let's consider the different ways to pick a pair of the same color. For the black shoes, she has 6 pairs, or 12 individual shoes. The number of ways to pick 2 black shoes is 12 P 2 = 12 * 11. For the brown shoes, with 3 pairs (6 individual shoes), it's 6 P 2 = 6 * 5. For the gray shoes, with 2 pairs (4 individual shoes), it's 4 P 2 = 4 * 3.
The sum of all the ways to pick a same-color pair is 12*11 + 6*5 + 4*3. So the probability is calculated by:
(12*11 + 6*5 + 4*3) / (22*21) = (132 + 30 + 12) / (462) = 174 / 462 = 1/3
Therefore, the probability that Sue picks two shoes of the same color and one is a left shoe and the other a right shoe is 1/3.