Two-thirds of the roads from A to B are at least 5 miles

Hi All,

We're told that 2/3 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. We're asked to randomly pick a road from A to B and then randomly pick a road from B to C and determine the probability that AT LEAST ONE of the roads you pick is at least 5 miles long. This question can be approached in a number of different ways. In most cases, when a question asks for the probability of 'at least one' outcome occurring, the fastest approach is to determine that the outcome does NOT occur at all - and then subtract that probability from the number 1 (as swerve has shown). The answers to this question are sufficiently 'spread out' that you don't actually have to do any math to get the solution - a little logic is all that's required.

To start, since 2/3 of the roads from A to B are at least 5 miles long, the probability of randomly choosing such a road from A to B OR B to C (OR both) MUST be GREATER than 2/3. Eliminate Answers A, B and C.

With the two remaining answers, we have a reasonable answer (re: Answer D: 75% of the time) and an unreasonable one (Answer E: 11/12, which is over 90% of the time). Since the probability from B to C is relatively small (just 1/4 = 25%), there's no way that the overall probability would be so close to 100%. Thus, there's only one answer that makes sense...

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

BTGmoderatorLU wrote:Source: GMAT Paper Tests

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

Solution:

The only way that you won’t pick a road that is at least 5 miles long is if you pick a road from A to B that is less than 5 miles long and you also pick a road from B to C that is less than 5 miles long. The probability of the former is ⅓, and the probability of the latter is 3/4. Therefore, the probability of picking no roads that are at least 5 miles long is 1/3 x 3/4 = 1/4. In other words, the probability of picking at least one road that is at least 5 miles long is 1 - 1/4 = 3/4.

Answer: D