Binomial Distributions

Transcription

1 Binomial Distributions Unit 9 Probability Distributions 5 minutes Warm Up! 1. Read the Gambler s Fallacy: Why We Expect to Strike It Rich After a Losing Streak. 2. This study shows that a part of the brain reacts to the impact of losing, and it might explain why people tend to increase their bets after losing when gambling. Explain how this type of split decision making may influence fighter pilots, firefighters, or police officers, as the article states. 3. Give other examples of this type of split decision making. 1

2 Source: Psychology Today, August 2002, p. 22. Binomial Distribution Many types of probability problems have only two outcomes or can be reduced to two outcomes. For example, when a coin is tossed, it can land heads or tails. when a baby is born, it will be either male or female. in a basketball game, a team either wins or loses. a true/false item can be answered in only two ways, true or There is a fixed number of n trials carried out. Other situations can be reduced to two outcomes. For example, a medical treatment can be classified as effective or ineffective, depending on the results. a person can be classified as having normal or abnormal blood pressure, depending on the measure of the blood pressure gauge. a multiple-choice question, even though there are four or five answer choices, can be classified as correct or incorrect. 2

3 Binomial Distribution A binomial random variable X is defined to the number of successes in n independent trials where the P( success ) is the constant p and the P( failure ) is the constant q. In the definition above notice the following conditions need to be satisfied for a binomial experiment: 1. There is a fixed number of n trials carried out. 2. Each trial can have inly two outces, is either a success or a failure. 3. The probability of success (p) remains constant from trial to trial. 4. The trials are independent, the outcome of a trial is not affected by the outcome of any other trial. Binomial Distribution If n is the number of trials of the binomial experiment and p is the probability of success, then the probability of x successes in n trials of the experiment is given by the probability function P(x), defined as follows: 3

4 Tossing Coins A coin is tossed 3 times. Find the probability of getting exactly two tails. This problem can be solved by looking at the sample space. There are three ways to get two tails. TTT, TTH, THT, HTT, HHT, HTH, THH, HHH But we can also do it using the binomial probability formula. Tossing Coins A coin is tossed 3 times. Find the probability of getting exactly two tails. This problem can be solved by looking at the sample space. There are three ways to get two tails. TTT, TTH, THT, HTT, HHT, HTH, THH, HHH But we can also do it using the binomial probability formula. 4

5 Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. 1:59 1:58 1:57 1:56 1:55 1:54 1:53 1:52 1:51 1:50 1:49 1:48 1:47 1:46 1:45 1:44 1:43 1:42 1:41 1:40 1:39 1:38 1:37 1:36 1:35 1:34 1:33 1:32 1:31 1:30 1:29 1:28 1:27 1:26 1:25 1:24 1:23 1:22 1:21 1:20 1:19 1:18 1:17 1:16 1:15 1:14 1:13 1:12 1:11 1:10 1:09 1:08 1:07 1:06 1:05 1:04 1:03 1:02 1:01 1:00 0:59 0:58 0:57 0:56 0:55 0:54 0:53 0:52 0:51 0:50 0:49 0:48 0:47 0:46 0:45 0:44 0:43 0:42 0:41 0:40 0:39 0:38 0:37 0:36 0:35 0:34 0:33 0:32 0:31 0:30 0:29 0:28 0:27 0:26 0:25 0:24 0:23 0:22 0:21 0:20 0:19 0:18 0:17 0:16 0:15 0:14 0:13 0:12 0:11 0:10 0:09 0:08 0:07 0:06 0:05 0:04 0:03 0:02 0:01 2:00 End Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. 5

6 Survey on Equipment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30%of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. 5 minutes Survey on Equipment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30%of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs =

7 Driving While Intoxicated A report from the Secretary of Health and Human Services stated that 70% of single vehicle traffic fatalities that occur at night on weekends involve an intoxicated driver. If a sample of 15 singlevehicle traffic fatalities that occur at night on a weekend is selected, find the probability that exactly 12 involve a driver who is intoxicated. 3 minutes Driving While Intoxicated A report from the Secretary of Health and Human Services stated that 70% of single vehicle traffic fatalities that occur at night on weekends involve an intoxicated driver. If a sample of 15 singlevehicle traffic fatalities that occur at night on a weekend is selected, find the probability that exactly 12 involve a driver who is intoxicated. Answer: 0.17 or 17% chance of happening 7

8 Can We Find Mean, Variance and Standard Deviation for Binomial Distribution Do you remember? Tossing a Coin A coin is tossed 4 times. Find the mean, variance, and standard deviation of the number of heads that will be obtained. Can We Find Mean, Variance and Standard Deviation for Binomial Distribution Do you remember? Tossing a Coin A coin is tossed 4 times. Find the mean, variance, and standard deviation of the number of heads that will be obtained. 8

9 Rolling a die Adie is rolled 360 times. Find the mean, variance, and standard deviation of the number of 4s that will be rolled. 1:59 1:58 1:57 1:56 1:55 1:54 1:53 1:52 1:51 1:50 1:49 1:48 1:47 1:46 1:45 1:44 1:43 1:42 1:41 1:40 1:39 1:38 1:37 1:36 1:35 1:34 1:33 1:32 1:31 1:30 1:29 1:28 1:27 1:26 1:25 1:24 1:23 1:22 1:21 1:20 1:19 1:18 1:17 1:16 1:15 1:14 1:13 1:12 1:11 1:10 1:09 1:08 1:07 1:06 1:05 1:04 1:03 1:02 1:01 1:00 0:59 0:58 0:57 0:56 0:55 0:54 0:53 0:52 0:51 0:50 0:49 0:48 0:47 0:46 0:45 0:44 0:43 0:42 0:41 0:40 0:39 0:38 0:37 0:36 0:35 0:34 0:33 0:32 0:31 0:30 0:29 0:28 0:27 0:26 0:25 0:24 0:23 0:22 0:21 0:20 0:19 0:18 0:17 0:16 0:15 0:14 0:13 0:12 0:11 0:10 0:09 0:08 0:07 0:06 0:05 0:04 0:03 0:02 0:01 2:00 End Rolling a die Adie is rolled 360 times. Find the mean, variance, and standard deviation of the number of 4s that will be rolled. 9

10 Likelihood of Twins The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. 1:59 1:58 1:57 1:56 1:55 1:54 1:53 1:52 1:51 1:50 1:49 1:48 1:47 1:46 1:45 1:44 1:43 1:42 1:41 1:40 1:39 1:38 1:37 1:36 1:35 1:34 1:33 1:32 1:31 1:30 1:29 1:28 1:27 1:26 1:25 1:24 1:23 1:22 1:21 1:20 1:19 1:18 1:17 1:16 1:15 1:14 1:13 1:12 1:11 1:10 1:09 1:08 1:07 1:06 1:05 1:04 1:03 1:02 1:01 1:00 0:59 0:58 0:57 0:56 0:55 0:54 0:53 0:52 0:51 0:50 0:49 0:48 0:47 0:46 0:45 0:44 0:43 0:42 0:41 0:40 0:39 0:38 0:37 0:36 0:35 0:34 0:33 0:32 0:31 0:30 0:29 0:28 0:27 0:26 0:25 0:24 0:23 0:22 0:21 0:20 0:19 0:18 0:17 0:16 0:15 0:14 0:13 0:12 0:11 0:10 0:09 0:08 0:07 0:06 0:05 0:04 0:03 0:02 0:01 2:00 End Likelihood of Twins The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. 10