Chapter 27
Probability is a quantity that expresses the chance, or likelihood, of an event. In other words, it measures how often an event will occur in a long series of repeated trials.
For events with countable outcomes, probability is defined by the following fraction:
This fraction assumes all outcomes are equally likely. If not, the math can be more complicated (more on this later).
As a simple illustration, rolling a die (singular for dice) has six possible outcomes: 1, 2, 3, 4, 5, and 6. The probability of rolling a “5” is 1/6, because the “5” corresponds to only one of those outcomes. The probability of rolling a prime number, though, is 3/6, which simplifies to 1/2, because in that case, three of the outcomes—2, 3, and 5—are considered successes.
Again, all the outcomes must be equally likely. For instance, you could say that the lottery has only two “outcomes”—win or lose—but that does not mean the probability of winning the lottery is 1/2. If you want to calculate the correct probability of winning the lottery, you must find all of the possible equally likely outcomes. In other words, you have to count up all the specific combinations of differently numbered balls in the lottery to determine the correct probability of winning the lottery.
In some problems, you will have to think carefully about how to break a situation down into equally likely outcomes. Consider the following problem:
If a fair coin is tossed three times, what is the probability that it will turn up heads exactly twice?
You may be tempted to say that there are four possibilities—no heads, 1 head, 2 heads, and 3 heads—and that the probability of 2 heads is thus 1/4. You would be wrong, though, because those four outcomes are not equally likely. You are much more likely to get 1 or 2 heads than to get all heads or all tails. Instead, you have to formulate equally likely outcomes in terms of the outcome of each flip:
HHH HHT HTH THH HTT THT
TTH TTT
If you have trouble formulating this list from scratch, you can use a counting tree, which breaks down possible outcomes step by step, with only one decision at each branch of the tree. An example is to the left.
These eight outcomes are equally likely, because the coin is equally likely to come up heads or tails at each flip. Three outcomes on this list (HHT, HTH, THH) have heads exactly twice, so the probability of exactly two heads is 3/8.
This result can also be written:
P(exactly 2 heads) = 3/8.