Vishnu88 wrote:Stuart, would you be kind enough to illustrate a probability problem with multiple scenarios. Just to get an idea of the one-minus method?
Sure! Let's use a very common GMAT probability question type: coin flips.
A fair coin is flipped 5 times. What's the probability that at least 1 of the 5 results is heads?
Whenever you see "at least" or "at most" in a question, you're dealing with multiple scenarios. Here, for example, we want "at least 1 head" out of 5 flips, so acceptable results are:
1 head and 4 tails;
2 heads and 3 tails;
3 heads and 2 tails;
4 heads and 1 tail; and
5 heads and 0 tails.
Since we want 1H4T OR 2H3T OR 3H2T OR 4H1T OR 5H01, one way to solve would be to figure out the probability of each scenario and ADD them together.
(Here's a very important rule to remember for probability and counting questions: to calculate MULTIPLE scenarios, MULTIPLY the individual ones; to calculate ALTERNATIVE scenarios, ADD the individual ones.)
However, if we're GMAT experts we notice that the only scenario that we DO NOT want is 0 heads and 5 tails. Since the sum of all probabilities is 1, we can use the "one minus" approach and set up the following equation:
Probability(what we want) = 1 - Probability(what we don't want)
Prob(at least 1 head) = 1 - Prob(0 heads)
0 heads is the same as 5 tails, and the probability of 5 tails is simply:
(1/2)(1/2)(1/2)(1/2)(1/2) = 1/32
So:
Prob(at least 1 head) = 1 - 1/32 = 31/32
