7 Probability. Copyright Cengage Learning. All rights reserved.

Suppose you have a coin that you think is not fair and you would like to determine the likelihood that heads will come up when it is tossed. You could estimate this likelihood by tossing the coin a large number of times and counting the number of times heads comes up. Suppose, for instance, that in 100 tosses of the coin, heads comes up 58 times. The fraction of times that heads comes up, 58 100 =.58, is the relative frequency, or estimated probability of heads coming up when the coin is tossed. 3

In other words, saying that the relative frequency of heads coming up is.58 is the same as saying that heads came up 58% of the time in your series of experiments. Now let s think about this example in terms of sample spaces and events. First of all, there is an experiment that has been repeated N = 100 times: Toss the coin and observe the side facing up. The sample space for this experiment is S = {H, T}. Also, there is an event E in which we are interested: the event that heads comes up, which is E = {H}. 4

The number of times E has occurred, or the frequency of E, is fr(e) = 58. The relative frequency of the event E is then 5

Relative Frequency When an experiment is performed a number of times, the relative frequency or estimated probability of an event E is the fraction of times that the event E occurs. If the experiment is performed N times and the event E occurs fr(e) times, then the relative frequency is given by Fraction of times E occurs 6

The number fr(e) is called the frequency of E. N, the number of times that the experiment is performed, is called the number of trials or the sample size. If E consists of a single outcome s, then we refer to P(E) as the relative frequency or estimated probability of the outcome s, and we write P(s). 7

Visualizing Relative Frequency The collection of the estimated probabilities of all the outcomes is the relative frequency distribution or estimated probability distribution. 8

Quick Examples 1. Experiment: Roll a pair of dice and add the numbers that face up. Event: E: The sum is 5. If the experiment is repeated 100 times and E occurs on 10 of the rolls, then the relative frequency of E is 9

2. If 10 rolls of a single die resulted in the outcomes 2, 1, 4, 4, 5, 6, 1, 2, 2, 1, then the associated relative frequency distribution is shown in the following table: 10

Example 1 Sales of Hybrid Vehicles In a survey of 250 hybrid Vehicles sold in the United States, 125 were Toyota Prii, 30 were Honda Civics, 20 were Toyota Camrys, 15 were Ford Escapes, and the rest were other makes. What is the relative frequency that a hybrid vehicle sold in the United States is not a Toyota Camry? Solution: The experiment consists of choosing a hybrid vehicle sold in the United States and determining its make. 11

Example 1 Solution cont d The sample space suggested by the information given is S = {Toyota Prius, Honda Civic, Toyota Camry, Ford Escape, Other} and we are interested in the event E = {Toyota Prius, Honda Civic, Ford Escape, Other}. The sample size is N = 250 of which 20 were Toyota Camrys. Thus, the frequency of E is fr(e) = 250 20 = 230 and the relative frequency of E is 12

Some Properties of Relative Frequency Distributions Let S = {s 1, s 2,..., s n } be a sample space and let P(s i ) be the relative frequency of the event {s i }. Then 1. 0 P(s i ) 1 2. P(s 1 ) + P(s 2 ) + + P(s n ) = 1 3. If E = {e 1, e 2,..., e r }, then P(E) = P(e 1 ) + P(e 2 ) + + P(e r ). 13

In words: 1. The relative frequency of each outcome is a number between 0 and 1 (inclusive). 2. The relative frequencies of all the outcomes add up to 1. 3. The relative frequency of an event E is the sum of the relative frequencies of the individual outcomes in E. 14

and Increasing Sample Size 15

and Increasing Sample Size A fair coin is one that is as likely to come up heads as it is to come up tails. In other words, we expect heads to come up 50% of the time if we toss such a coin many times. Put more precisely, we expect the relative frequency to approach.5 as the number of trials gets larger. Figure 4 shows how the relative frequency behaved for one sequence of coin tosses. Figure 4 16

and Increasing Sample Size For each N we have plotted what fraction of times the coin came up heads in the first N tosses. Notice that the relative frequency graph meanders as N increases, sometimes getting closer to.5, and sometimes drifting away again. However, the graph tends to meander within smaller and smaller distances of.5 as N increases. 17

and Increasing Sample Size In general, this is how relative frequency seems to behave; as N gets large, the relative frequency appears to approach some fixed value. Some refer to this value as the actual probability, whereas others point out that there are difficulties with this notion. 18