If you were dealing with only 2 or 3 days, then listing out the possibilities might be a good approach. But because we have 5 days, listing is definitely time-consuming and error-prone.
Here's a couple of other approaches.
For any question that involves a 50/50 probability between 2 events (rain/no rain, head/tail, etc), you can use the formula nCk/2^n where "n" is the number of times the event is being exercised and "k" represents desirable outcomes. "nCK" represents the number of ways the desirable event can occur while "2^n" is the total number of possibilities (so this is just the basic probability formula). The "2" comes from the fact that it is a binary event: rain/no rain. (So, if it were a die toss, we would use 6^n in the denominator.)
Here, we want rain on exactly 3 of the 5 days. So, n equals 5. We have: 5C3/2^5 = 5/16
You can also use Pascal's triangle on these problems:
.............................1 (n = 0)
..........................1.....1 (n = 1)
......................1......2........1 (n = 2)
..................1......3.......3.........1 (n = 3)
..............1......4.......6........4..........1 (n = 4)
..........1.......5.....10.....10.........5..........1 (n = 5)
.........(0)....(1)....(2)....(3).........(4).......(5) [5th row k values]
or
........(5).....(4)....(3)....(2).........(1).......(0) [5th row k values]
The triangle has many symmetrical properties that make it easy to remember. For example, each digit in a row (except for 1) is the sum of the digits directly above to the left and directly above to the right. Most GMAT questions will involve 3 or 5 tosses (or in this case, days). (So, you could just memorize the n =3 and n = 5 rows). Because there are 5 days, we go to the sixth row where n = 5. We want to know how it will rain on exactly three days. "10" is the third digit in the n = 5 row (whether you start counting from the left or the right). The sum of the digits in that row is 32. Therefore, the probability is 10/32 = 5/16.
EDIT: when counting, you need to start counting at zero. Looking at the bottom row, if we start from either the left or right, the "1" represents 1/32 chance of getting 0 rainy days (or ALL rainy days, which is just 0 non-rainy days); the "5" represents the chances of getting 1 rainy day out of 5 (or 1 non-rainy day or 1 head, etc). (So, the chances of getting exactly 1 rainy day are 5/32.) So, when I said "10" was the "third" digit, I should have made it clear that I meant zeroth, first, second, and third digit!
The position of each digit in a row you're looking at is just the k value you're interested in counting. After numerous attempts, I have edited the triangle to reflect this.
Last edited by
Testluv on Mon Dec 21, 2009 11:59 pm, edited 4 times in total.
Kaplan Teacher in Toronto
