Question

You aren't sure if your bridesmaids dress is yellow or blue. According to fake industry statistics 80% of dresses are blue, while 20% are yellow, when looking at a yellow dress, 75% of people correctly believe the dress is yellow, when looking at a blue dress, 90% of people correctly believe the dress is blue. Polling your friends, 10 out of 20 believe the dress is yellow. What is more likely the dress is blue or yellow?

Answer 1

Let Y, B be the events denoting that the dress is yellow and blue respectively.

Let BY, BB be the events denoting that the people believe that the dress is yellow and blue respectively.

P(B) = 0.8

P(Y) = 0.2

P(BY | Y) = 0.75 =>  P(BB | Y) = 1 - P(BY | Y) = 1 - 0.75 = 0.25

P(BB | B) = 0.9

P(BY) = 10/20 = 0.5

P(BB) = 1 - P(BY) = 1 - 0.5 = 0.5

By Bayes theorem,

P(Y | BY) = P(BY | Y) * P(Y) / P(BY) = 0.75 * 0.2 / 0.5 = 0.3

P(Y | BB) = P(BB | Y) * P(Y) / P(BB) = 0.25 * 0.2 / 0.5 = 0.1

Thus, given the people believe the dress is yellow or blue, the likelihood of the dress is yellow is less than 0.5. So, the more likely dress is blue.