A closet contains 13 different pairs of gloves. The 26 gloves are randomly arranged into 13 pairs.
(a) Find the probability that all left-hand gloves are paired with right-hand gloves (not necessarily matching).
(b) Find the probability that all of the gloves are arranged so that each left-hand glove is paired with its matching right-hand glove.
To find the probability of events in this scenario, we need to calculate the total number of possible outcomes and the number of favorable outcomes.
Let's start with part (a):
(a) Probability that all left-hand gloves are paired with right-hand gloves (not necessarily matching):
To determine the probability of this event, we need to find the number of favorable outcomes and the total number of possible outcomes.
Favorable outcomes: For each left-hand glove, there are 13 possible right-hand gloves to pair with.
To calculate the favorable outcomes, we can use the concept of permutations. We need to figure out how many ways we can arrange the 26 gloves so that each left-hand glove is paired with any of the possible 13 right-hand gloves.
The number of favorable outcomes can be calculated as: 13! (arrange the 13 pairing possibilities for each of the 13 left-hand gloves)
Total outcomes: There are 26 gloves in total, so to calculate the total number of outcomes, we can use the concept of permutations again.
The total number of outcomes can be calculated as: 26!
Using these numbers, we can find the probability as the ratio of favorable outcomes to the total outcomes:
Probability (a) = Favorable outcomes / Total outcomes = 13! / 26!
Now, let's move on to part (b):
(b) Probability that all of the gloves are arranged so that each left-hand glove is paired with its matching right-hand glove:
In this case, for each left-hand glove, there is only one specific right-hand glove that matches it.
Favorable outcomes: Each left-hand glove has one matching right-hand glove. So the number of favorable outcomes will be 1 for each left-hand glove, resulting in a total of 1^13 (since there are 13 pairs of gloves).
Total outcomes: The total number of outcomes can be calculated using permutations. Since there are 26 gloves total, the total outcomes will be 26!
Using these numbers, we can calculate the probability as:
Probability (b) = Favorable outcomes / Total outcomes = (1^13) / 26!
Now you can calculate the probabilities for both (a) and (b) using the formulas provided.