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British and American spelling are `rigour' and `rigor', 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% of the English-speaking men at the hotel are British and 60% are Americans, what is the probability that the writer is British?

Answer :

Answer:

45.45%

Step-by-step explanation:

Given:

British spelling as "rigour"

here Probability of selecting vowel , P(V/B) =  [tex]\frac{\textup{Number of vowels}}{\textup{Total letters}}[/tex]

or

Probability of selecting vowel , P(V/B) =  [tex]\frac{\textup{3}}{\textup{6}}[/tex] = 0.5

and,

American spelling as "rigor"

here Probability of selecting vowel , P(V/A) =  [tex]\frac{\textup{Number of vowels}}{\textup{Total letters}}[/tex]

or

Probability of selecting vowel , P(V/A) = [tex]\frac{\textup{2}}{\textup{5}}[/tex] = 0.4

Probability of English-speaking men at hotel are British P(B) = 40% = 0.4

Probability of English-speaking men at hotel are American P(A) = 60% = 0.6

Now using Baye's theorem

the probability that the writer is British P(B/V) = [tex]\frac{P(B)\times P(V/E)}{P(B)\times P(V/E)+P(A)\times P(V/A)}[/tex]

or

the probability that the writer is British P(B/V) = [tex]\frac{0.4\times0.5}{0.4\times0.5+0.6\times0.4}[/tex]

or

the probability that the writer is British P(B/V) = [tex]\frac{0.2}{0.44}[/tex]

or

the probability that the writer is British P(B/V) = 0.4545 = 45.45%

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