CHAPTER 16 Basic Concepts of Statistics, Counting, and Probability


The probability that an event will occur is a number between 0 and 1, often written as a fraction, that indicates how likely it is that the event will happen. For example, if you spin the spinner below, there are six possible outcomes. Since each region The is the same size, it is equally likely that the spinner will stop in any of the six regions. There is 1 chance in 6 that it will stop in the region marked 2. So we say that the probability of spinning a 2 is one-sixth and write . Since 2 is the only even number on the spinner, we could also say . There are 5 chances in 6 that the spinner will land in a region with an odd number in it, so 

Key Fact O3

If E is any event, the probability that E will occur is given by

assuming that all of the possible outcomes are equally likely.

The examples cited below and in the next Key Fact all refer to the spinner shown above.

Key Fact O4

Let E be an event and P(E ) the probability it will occur.

• If E is impossible (such as getting a number greater than 15), P(E ) = 0.

• If it is certain that E will occur (such as getting a prime number), P(E ) = 1.

• In all cases 0 P(E ) 1.

• The probability that event E will not occur is 1 − P(E ).

• If two or more events are mutually exclusive and constitute all the outcomes, the sum of their probabilities is 1. (For example, P(even) + P(odd) =  = 1.)

• The more likely that an event will occur, the higher its probability (the closer to 1 it is); the less likely that an event will occur, the lower its probability (the closer to 0 it is).

Even though probability is defined as a fraction, we can write probabilities as decimals or percents, as well. Instead of writing P(E) = , we can write P(E ) = .50 or P(E ) = 50%.

EXAMPLE 9: An integer between 100 and 999, inclusive, is chosen at random. What is the probability that all three digits of the number are different odd numbers?

If E represents the event of picking a number with three different odd digits, then

In Example 4, we calculated that the numerator of this fraction is 60. The denominator is 900, which you could get by deleting the 99 integers from 1 to 99 from the 999 integers from 1 to 999 or by using the counting principle:

Key Fact O5

If an experiment is repeated two (or more) times, the probability that an event occurs and then a second event occurs is the product of the probabilities that each event occurs.

EXAMPLE 10: If the spinner discussed above is spun three times, what is the probability that it will stop in the region marked 2 each time?

Each time the spinner is spun, . Therefore,

P(3 2s in a row)=

 P(2 on the 1st spin) x P(2 on the 2nd spin)x P(2 on the 3rd spin)=