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 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