Bunuel wrote:

Two-thirds of the roads from A to B are at least 5 miles long, and 1/4 of the roads from B to C are at least 5 miles long. If you randomly pick a road from A to B and then randomly pick a road from B to C, what is the probability that at least one of the roads you pick is at least 5 miles long?

(A) 1/6

(B) 1/4

(C) 2/3

(D) 3/4

(E) 11/12

I think D.

Probability of "at least one of the roads'" being at least 5 miles long is the complement of the event never occurring; that is, find probability of none for each event, then multiply. Subtract from one. (Because P(A) + P(not A) = 1.)

Probability of "none" =

For A to B: (1 - 2/3) = 1/3

For B to C: (1 - 1/4) = 3/4

P(none) = \((\frac{1}{3}*\frac{3}{4})=\frac{3}{12}=\frac{1}{4}\)

1 - P(none) = (1 - \(\frac{1}{4})=\frac{3}{4}\)

Answer D, IMO

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