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 that there is at least one matching pair among the six socks taken at random and without replacement from eight pairs of socks is \(1 - \frac{{C(8, 6) \times 2^6}}{{C(16, 6)}}\).

Step-by-step solution

Calculate total sample space

Calculate the total ways of choosing 6 socks from 16. This can be done using the combination \(C(n, r)\) formula where \(n = 16\) and \(r = 6\). The formula is \(C(16, 6)\).

Calculate sample space of not having a matching pair

Calculate the ways of choosing 6 socks from 16 such that there is no pair. Here we are choosing one sock from each pair, hence it is equal to \(C(8, 6)\). But each sock selected can be any one from the 16 socks. Hence, the total ways of choosing 6 socks such that there is no matching pair is \(C(8, 6) \times 2^6\).

Calculate the probability

Now, the probability that there are no matching pairs among the 6 chosen socks is the ratio of the ways of choosing 6 socks without a matching pair to the total ways of choosing 6 socks, i.e. \(P(\text{{no pair}}) = \frac{{C(8, 6) \times 2^6}}{{C(16, 6)}}\). But, we need the probability that there is at least a pair. Hence, \(P(\text{{at least one pair}}) = 1 - P(\text{{no pair}})\).