Madelaine88 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

We are given that 2/3 of the roads from A to B are at least 5 miles long (which means that 1/3 of the roads from A to B are less than 5 miles long) and 1/4 of the roads from B to C are at least 5 miles long (which means that 3/4 of the roads from A to B are less than 5 miles long). We need to determine the probability that when picking a road from A to B and B to C, at least one of the roads is at least 5 miles long.

We can use the following formula:

P(selecting at least 1 road that is at least 5 miles long) + P(selecting no roads that are at least 5 miles long) = 1

P(selecting at least 1 road that is 5 miles long) = 1 - P(selecting no roads that are at least 5 miles long)

P(selecting at least 1 road that is 5 miles long) = 1 - P(selecting all roads that are less than 5 miles long)

Thus, if we can determine the probability of selecting all roads that are less than 5 miles long, we’ll quickly be able to calculate the probability of selecting at least 1 road that is at least 5 miles long.

The probability of selecting all roads that are less than 5 miles long is: 1/3 x 3/4 = 1/4

Thus, the probability of selecting at least 1 road that is at least 5 miles long is: 1 - 1/4 = 3/4

Answer: D

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