English and American spellings are colour and color, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the hotel are English and 60 percent are Americans, what is the probability that the writer is an Englishman?

The probability is 5/11

Step-by-step explanation:

Let's call V the event that the letter taken at random is a vowel.

Let's call E the event that the man is English and A to the event that the man is American.

If 40 percent of the English-speaking men at the hotel are English means P (E) = 0.40 and 60 percent are Americans means P (A) = 0.60

In ''color'' we have 2 vowels out of 5 letters so P (V/A) = 2/5

In ''colour'' we have 3 vowels out of 6 letters so P (V/E) = 3/6=1/2

P (E/V) = P (E∩ V) / P (V)

P (V) = P (V|E) P (E) + P (V|A) P (A)

P (V) = (1/2) 0.40 + (2/5) 0.60=0.44

P (E∩V) = P (V|E) P (E) = (1/2) 0.40=0.20

P (E/V) = 0.20/0.44=0.45454545=5/11