Probability and Statistics Copyright Cengage Learning. All rights reserved.
14.3 Binomial Probability Copyright Cengage Learning. All rights reserved.
Objectives Binomial Probability The Binomial Distribution 3
Binomial Probability A coin is weighted so that the probability of heads is 0.6. What is the probability of getting exactly two heads in five tosses of this coin? Since the tosses are independent, the probability of getting two heads followed by three tails is 0.6 0.6 0.4 0.4 0.4 = (0.6) 2 (0.4) 3 4
Binomial Probability But this is not the only way we can get exactly two heads. The two heads can occur, for example, on the second toss and the last toss. In this case the probability is 0.4 0.6 0.4 0.4 0.6 = (0.6) 2 (0.4) 3 5
Binomial Probability In fact, the two heads could occur on any two of the five tosses. Thus there are C (5, 2) ways in which this can happen, each with probability (0.6) 2 (0.4) 3. It follows that P (exactly 2 heads in 5 tosses) = C(5, 2)(0.6) 2 (0.4) 3 0.023 The probability that we have just calculated is an example of a binomial probability. In general, a binomial experiment is one in which there are two outcomes, which are called success and failure. 6
Binomial Probability In the coin-tossing experiment described above, success is getting heads, and failure is getting tails. The following box tells us how to calculate the probabilities associated with binomial experiments when we perform them many times. 7
Example 1 Binomial Probability A fair die is rolled 10 times. Find the probability of each event. (a) Exactly 2 sixes. (b) At most 1 six. (c) At least 2 sixes. Solution: Let s call getting a six success and not getting a six failure. So P (success) = and P (failure) = Since each roll of the die is independent of the other rolls, we can use the formula for binomial probability with n = 10 and p =. 8
Example 1 Solution cont d (a) P (exactly 2 sixes) = 0.29 (b) The statement at most 1 six means 0 sixes or 1 six. So P (At most one six) = P (0 sixes or 1 six) = P (0 sixes) + P (1 six) = C(10,0) + C(10,1) Meaning of at most P(A or B) = P(A) + P(B) Binomial probability 9
Example 1 Solution cont d 0.1615 + 0.3230 0.4845 (c) The statement at least two sixes means two or more sixes. Instead of adding the probabilities of getting 2, 3, 4, 5, 6, 7, 8, 9, or 10 sixes (which is a lot of work), it s easier to find the probability of the complement of this event. 10
Example 1 Solution cont d The complement of the event two or more sixes is 0 or 1 six. So P (two or more sixes) = 1 P(0 or 1 six) = 1 0.4845 P(E) = 1 P(Eʹ ) From part (b) = 0.5155 11
The Binomial Distribution 12
The Binomial Distribution We can describe how the probabilities of an experiment are distributed among all the outcomes of an experiment by making a table of values. The function that assigns to each outcome its corresponding probability is called a probability distribution. A bar graph of a probability distribution in which the width of each bar is 1 is called a probability histogram. The next example illustrates these concepts. 13
Example 3 Probability Distributions Make a table of the probability distribution for the experiment of rolling a fair die and observing the number of dots. Draw a histogram of the distribution. Solution: When rolling a fair die each face has probability 1/6 of showing. 14
Example 3 Solution cont d The probability distribution is shown in the following table. To draw a histogram, we draw bars of width 1 and height corresponding to each outcome. Probability Histogram Probability Distribution 15
The Binomial Distribution A probability distribution in which all outcomes have the same probability is called a uniform distribution. The rolling-a-die experiment in Example 3 is a uniform distribution. The probability distribution of a binomial experiment is called a binomial distribution. 16
Example 4 A Binomial Distribution A fair coin is tossed eight times, and the number of heads is observed. Make a table of the probability distribution, and draw a histogram. What is the number of heads that is most likely to show up? Solution: This is a binomial experiment with n = 8 and p =, so 1 p = as well. We need to calculate the probability of getting 0 heads, 1 head, 2 heads, 3 heads, and so on. For example, to calculate the probability of 3 heads we have P (3 heads) = C(8, 3) 17
Example 4 Solution cont d The other entries in the following table are calculated similarly. We draw the histogram by making a bar for each outcome with width 1 and height equal to the corresponding probability. Probability Histogram Probability Distribution 18