Probability of roads

"at least 5 miles long" = X

From A to B:
Prob(X_ab) = 2/3
Prob(not X_ab) = 1/3

From B to C:
Prob(X_bc) = 1/4
Prob(not X_bc) = 3/4

Prob of at least 1 X from A to C =
=1 - Prob(not X_ac)
=1 - Prob(not X_ab)*Prob(not X_bc)
=1 - (1/3)*(3/4)
=1- 1/4
=3/4

pakaskwa 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

pakaskwa 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

OA is D. But I don't really understand the calculation. Can somebody help me to explain?

2/3rd of the roads are A to B are ≥5miles
Therefore 1/3 of the roads are <5miles
¼ th of the roads from C to D are ≥ 5miles
Therefore 3/4th of the roads are <5miles
so probabilility of both the roads <5miles =(1/3)*(3/4)=1/4
hence probabilility of one of the roads ≥5miles=1-1/4=3/4
so D is the correct option .