Question

Exercise 2.38. 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: SOME DOGS ARE BROWN (a) Find the probability that the chosen letter is R. (b) Let X denote the length of the chosen word. Determine the probability mass function of X. (c) For each possible value k of X determine the conditional probability P(X k|X 3) Hint. The decomposition idea works just as well for conditional probabilities: if (B\, , Bn} is a partition of 2, then P(A| D) = PABk|D). k=1 (d) Determine the conditional probability P(the chosen letter is R | X > 3). (e) Given that the chosen letter is R, what is the probability that the chosen word was BROWN?

Answer 1

BROWN ARE SOME DOGS a)P(ctosen leter is R) = P(chosen uword is ARE) x P(chosen lete is R) P (chosen coord is BROWN) x P Cchasen leter fsR) 5 6) ) 4 %=3 4 K=4 % =5 oreauoise P(X-3 1X3) P(x-4, X>3) P (X3) P(X- P(X>3) P (x4 x3) (4) (34) P (x-5 X3) P(x-5,x>3) P(x-5) P(x>3) P(x73) (V4) P(x-k x3) or k4,5 = 0

d)Pchoseh leten isRI X>3) =P(chosenlate is R, X>3) P(x>3) 15

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