A pet supplier has a stock of parakeets of which 20 are blue

A pet supplier has a stock of parakeets of which 20% are blue parakeets. A pet store orders 6 parakeets from this supplier. If the supplier selects the parakeets at random, what is the chance that the pet store gets exactly one blue parakeet? Write your answer as a proportion and round to 3 decimal places.

Solution

These types of problems are deceptive. The probability of a blue parakeet is 0.10 and the probability of a non-blue one is 0.90, so we can calculate probabilities of the different possible combinations: no blues: 0.90^3 = 0.729 1 blue: 0.90^2 * .10 = 0.081 2 blues: 0.90 * 0.10^2 = 0.009 3 blues: 0.10^3 = 0.001 It should be obvious there\'s something wrong here, because these don\'t add up to 1. What\'s wrong is that, if we imagine the birds being picked out one at a time, we realize that there are actually three ways to pick one blue and two non-blue: B, N, N N, B, N N, N, B and similarly three ways to pick two blue and one non-blue. (There\'s only one way to pick all blue, or all non-blue.) So the chances of picking one or two blues are actually three times as great, and in particular the chance of picking one blue is 0.081 * 3 = 0.243 (answer B).