.Probability problems that deal with multiple events usually involve two primary operations: multiplication and addition. The key to understanding probability is to understand when you must multiply and when you must add.
Assume that X and Y are independent events. Two events are said to be independent if the likelihood of one occurring does not depend on the likelihood of the other occurring. To determine the probability that event X AND event Y will both occur, MULTIPLY the two probabilities together. Note that the events must be independent for this to work.
For example:
What is the probability that a fair coin flipped twice will land on heads both times?
This is an “AND” problem, because it is asking for the probability that the coin will land on heads on the first flip AND on the second flip.
Notice that there's a sequence of events here: the first flip happens, then the second flip happens. Fortunately, many “AND” problems involve a sequence like this one, or you can pretend that there is one. For instance, if you are asked about a coin flip and a roll of a die, you can pretend that the coin flip comes before the die roll.
Either way, you can imagine the sequence of events as a series of “forks in the road.” In the case of the two coin flips, the first fork is the first flip, and you want to take the “heads” path (not the “tails” one). Then you come to the second fork, and again, you want to take the "heads" path.
The probability that the coin will land on heads on the first flip is 1/2. The probability that the coin will land on heads on the second flip is 1/2. These events are independent of each other.
Therefore, to determine the probability that the coin will land on heads on both flips, multiply the probabilities:
.
Note that the probability of having BOTH flips come up heads (1/4) is less than the probability of having just one flip come up heads (1/2). This should make intuitive sense. If you define success in a more constrained way (e.g., “to win, BOTH this AND that have to happen”), then the probability of success will be lower. In other words, you have to take the correct path at the first fork AND at the second fork. That's harder to do.
The operation of multiplication should also make sense. Typical probabilities are fractions between 0 and 1. When you multiply together two such fractions, you get a smaller result, which means a lower probability.
Now assume that X and Y are mutually exclusive events (meaning that the two events cannot both occur). To determine the probability that event X OR event Y will occur, ADD the two probabilities together.
For example:
What is the probability that a fair die rolled once will land on either 4 or 5?
This is an “OR” problem, because it is asking for the probability that the die will land on either 4 OR 5. The probability that the die will land on 4 is 1/6. The probability that the die will land on 5 is 1/6. The two outcomes are mutually exclusive: the die cannot land on BOTH 4 and 5 at the same time.
Therefore, to find the probability that the die will land on either 4 or 5, add the probabilities:
.
In this case, you don't have a sequence of events. There is just one roll of the die. The two events you care about are two different possible outcomes from that same roll.
In other words, you don't have two successive “forks in the road” as you did before. Now you have just one fork—one roll of the die—with six different paths coming out of it, one for each possible number you can roll (1, 2, 3, 4, 5, or 6). The two paths you care about (rolling a 4 or rolling a 5) are two possible events resulting from the same fork in the road. This situation is common for “OR” problems.
Note that the probability of having the die come up either 4 or 5 (1/3) is greater than the probability of a 4 by itself (1/6) or of a 5 by itself (1/6). This should make intuitive sense. If you define success in a less constrained way (e.g., “I can win EITHER this way OR that way”), then the probability of success will be higher. The operation of addition should also make sense. Typical probabilities are fractions between 0 and 1. When you add together two such fractions, you get a larger result, which means a higher probability.
All that said, the majority of probability questions on the GRE are of the “AND” variety. When in doubt, multiply. Most GRE probability problems just want you to multiply two or three fractions together.
If a die is rolled twice, what is the probability that it will land on an even number both times?
Eight runners in a race are equally likely to win the race. What is the probability that the race will be won by the runner in lane 1 OR the runner in lane 8?
Advanced note: For adding “OR” probabillities, up until now it has been assumed that the events are mutually exclusive (meaning that both events cannot occur). What happens if the events are not mutually exclusive?
If that is the case, and you simply add the probabilities, you will be double-counting the instances when both events occur. Thus, you must subtract out the probability that both events occur.
If events X and Y are not mutually exclusive, then P(X OR Y) = P(X) + P(Y) − P(X AND Y). For example:
Suppose a box contains 20 balls. Ten balls are white and marked with the integers 1–10. The other 10 balls are red and marked with the integers 11–20. If one ball is selected, what is the probability that the ball will be white OR will be marked with an even number?
Because half the balls are white and half are marked with an even number P(white) + P(even) would give you
, which equals 1. This is incorrect! You must subtract out the probability that the ball is both white AND marked with an even number. There are 5 such balls out of 20. Thus, the correct answer is: P(white or even) = P(white) + P(even) − P(white AND even) =
.
A fair die is rolled and a fair coin is flipped. What is the probability that either the die will come up 2 or 3, OR the coin will land heads up?