AAPL wrote:John and Amanda stand at opposite ends of a straight road and start running towards each other at the same moment. Their rates are randomly selected in advance so that John runs at a constant rate of 3, 4, 5, or 6 miles per hour and Amanda runs at a constant rate of 4, 5, 6, or 7 miles per hour. What is the probability that John has traveled farther than Amanda by the time they meet?
A. 3/16
B. 5/16
C. 3/8
D. 1/2
E. 13/16
John will travel farther than Amanda if John is traveling faster than Amanda. That is, John's rate must be greater than Amanda's rate. Thus, John must travel either 5 or 6 miles per hour in order to have a chance to be faster than Amanda.
If John travels 5 mph (there is a 1/4 chance he would do that), then Amanda has to travel 4 mph (there is a 1/4 chance she would do that). Thus, the probability is 1/4 x 1/4 = 1/16.
If John travels 6 mph (there is a 1/4 chance he would do that), then Amanda has to travel either 4 or 5 mph (there is a 2/4 = 1/2 chance she would do that). Thus, the probability is 1/4 x 1/2 = 1/8.
Therefore, the overall probability that John will travel farther than Amanda is 1/16 + 1/8 = 3/16.
Alternate Solution:
Since there are 4 possibilities for the rates of John and Amanda each, there are 4 x 4 = 16 possible ways to assign rates for them.
Of these 16 possibilities, the only way John travels further than Amanda is if John = 6 mph and Amanda = 5 mph; John = 6 mph and Amanda = 4 mph; or if John = 5 mph and Amanda = 4 mph. Since there are 3 favorable outcomes in a total of 16 outcomes, the probability is 3/16.
Answer: A
