Question

3. We 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):

THE QUICK BROWN FOX JUMPED OVER THE GATE

What is the probability that the chosen letter is O, given that the chosen word has at least 4 letters?

Answer 1

Here, there are 8 words & one word is chosen uniformly at random random .Hence , probability of selecting each word is 1/8.

numbers of words of at least 4 letters=5

total words=8

P(words with at least 4 letters) = 5/8=0.625

Note that there are 2 words with at least 4 letters which contain letter O i.e brown,over

P(chosen letters o)=P(brown)*P(choosing O in Brown)+P(choosing over)*P(choosing O in Over) = 1/8*1/5 + 1/8*1/4=0.05625

sp, P(that the chosen letter is O | the chosen word has at least 4 letters) = P(chosen letter O and words with alleast 4 letters)/P(words with at least 4 letters) = 0.05625/0.625 = 0.09