Manhattan GMAT Challenge Problem of the Week – 5 Aug 2010
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As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!
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?
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:
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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