Question

A drawer contains eight different pairs of socks. If six socks are taken at random and without replacement, compute the probability that there is at least one matching pair among these six socks. Hint: Compute the probability that there is not a matching pair.

Short answer

The probability of getting at least one matching pair of socks among the six selected at random is approximately 0.590.

Step-by-step solution

Identify Total Possible Outcomes

Let's first calculate the total number of possible ways of selecting six socks from 16 socks. This can be done by computing \(_{16}C_{6}\). In general, \(_{n}C_{r}\) is a way to choose 'r' objects from 'n' possibilities. So total possible outcomes = \(_{16}C_{6}=4368.

Calculate Ways of Not Getting a Pair

Next, we need to compute the total number of ways to choose six socks without getting a pair. This is equivalent to picking one sock from each of the first six pairs and none from the other two pairs. This can be done by \(_{8}C_{6}\) * 2^6 ways. So, no pair outcomes = \(_{8}C_{6}\) * 2^6 = 1792.

Compute Probability of not getting a Pair

The probability of an event is given by the total number of favorable outcomes divided by the total number of possible outcomes. So, the probability of not getting a pair, P(not pair) = no pair outcomes / total possible outcomes = 1792 / 4368 = 0.410 .

Calculate Probability of Getting at Least One Pair

To find the probability of getting at least one pair, we subtract the probability of not getting a pair from 1, because the sum of the probabilities of all possible outcomes equals 1. So, probability of at least one pair, P(pair) = 1 - P(not pair) = 1 - 0.410 = 0.590.