Question

Choose one of the words in the following sentence uniformly at random and then choose one of the letters of that word, again uniformly at random:

SOME DOGS ARE BROWN

Given that the chosen letter is R, what is the probability that the chosen word was BROWN?

Answer 1

There are 4 words and these are equally likely so probability of selecting any word is 1/4. That is

P(SOME) = P(DOGS) =P(ARE)=P(BROWN) = 1/4

Now word SOME has no letter R so

P(R | SOME) = 0

Word DOGS has no letter R so

P(DOGS) = 0

Word ARE has one letter R out of three so

P(ARE) = 1/3

Word BROWN has one letter R out of five so

P(BROWN) = 1/5

By the Bayes' theorem the probability that the chosen word was BROWN given that chosen letter is R will be

Answer: 0.375