Problem Solving

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 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.

In these types of probability questions, it's often easiest to calculate the probability of what you DON'T WANT (and then subtract that from 1 to get the probability of what you DO want); Mitch's approach showcased that math. You can calculate the various ways to 'fit' what you DO want though - although it will be a bit more work (and in certain questions, it would be a lot more work).

There would be 3 ways for AT LEAST one of the two roads being at least 5 miles long:
1) First road IS, second road ISN'T
2) First road ISN'T, second road IS
3) Both roads ARE

The individual probabilities of those three events are:
1) (2/3)(3/4) = 6/12
2) (1/3)(1/4) = 1/12
3) (2/3)(1/4) = 2/12
Total probability = 6/12 + 1 /12 + 2/12 = 9/12 = 3/4

Final Answer: D

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

BTGmoderatorDC 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

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 â…“, 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