A study of vehicle flow characteristics on acceleration lanes (i.e., merging ram

Question

A study of vehicle flow characteristics on acceleration lanes (i.e., merging ramps) at a major freeway in Israel found that one out of every six vehicles uses less than one-third of the acceleration lane before merging into traffic (Journal of Transportation Engineering, Nov. 1985). Suppose we monitor the location of the merge for the next five vehicles that enter the acceleration lane.

a. What is the probability that none of the vehicles will use less than one-third of the acceleration lane?

b. What is the probability that exactly two of the vehicles will use less than one-third of the acceleration lane?

Explanation / Answer

a. The probability is 5/6 that any given vehicle will not use less than 1/3 of the lane. The probability that 5 out of 5 will not do so is therefore
(5/6)^5 = 3125 / 7776 = about 40.2%

b. The probability of any specific sequence of five actions, two of them being "using less than one-third of the lane" (probability 1/6) and the other three being "using at least one-third of the lane" (probability 5/6) is
(1/6)^2 * (5/6)^3

There are 5C2 = 10 different possible sequences, each equally probable, so the total probability of any of these sequences is
10 * (1/6)^2 * (5/6)^3
= 2 * 5^4 / 6^5 = 1250 / 7776 = 625 / 3888 = about 16%

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