Today’s GMAT challenge question comes from ManhattanGMAT.  To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:

Problem

Every trading day, the price of CF Corp stock either goes up by $1 or goes down by $1 with equal likelihood. At the end of 5 trading days, what is the probability that the price of CF Corp stock is up by exactly $3 from its initial price?

(A) 1/16

(B) 1/8

(C) 5/32

(D) 9/32

(E) 3/8

Solution

The daily change in CF Corp’s stock price can be compared to a coin flip. Heads – the price goes up by $1. Tails – the price goes down by $1. Moreover, the coin is “fair”: that is, each daily outcome is equally possible (meaning that the chance of heads is 50%, and the chance of tails is 50%). This also means that any particular sequence of flips is equally probable. As a result, our probability calculation is simplified. We just count the 5-day sequences that give us a $3 increase, then we compare that number to the total number of 5-day sequences.

To go up exactly $3, we need exactly 4 “up” days (heads) and 1 “down” day (tails). We don’t care about the order in which these days come. So we need to count the possible arrangements of 4 heads and 1 tails. We can simply list these out:

HHHHT

HHHTH

HHTHH

HTHHH

THHHH

The only question is what day of the week the “down” day falls on, so there are 5 possibilities for a $3 increase. Alternatively, we can use the combinations or anagrams method to calculate (5!)/(4!) = 5.

Now, we need to count all the possible 5-day sequences of flips. Since each day can have 2 outcomes (H or T), we have 2 × 2 × 2 × 2 × 2 = 32 total possible outcomes over 5 days.

Finally, to compute the probability of a $3 increase over 5 days, we divide 5 successful outcomes by 32 total possibilities (of equal weight) to get 5/32.

The correct answer is (C).

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