Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20! How are each of the following the same? - Toss a coin 5 times, count the number of heads. - Spin a roulette wheel 8 times, record the number of times the ball lands in a red slot. - Randomly sample 100 babies born in BC on a given date, count the number of males born. - Drawing a card from at standard deck and replacing it four times and observing the number of aces you get. How are these different from: - Tossing a coin until you count 5 heads. - Spinning a roulette wheel until the ball lands in the red slot 8 times. - Randomly sample 100 babies born in BC on a given date, count the number of males born in each ethnicity: Asian, First nations, Caucasian, Afro-American. - Drawing four cards from a standard deck and counting the number of aces you get.
Binomial Setting Characteristics of a Binomial Setting: - Binary - only two possible outcomes ( success or failure ) - Independent each trial of the same chance process is independent - Number the number of trials n of the chance process is fixed in advance - Success on each trial, the probability of success must be the same Definition: Binomial random variable and binomial distribution The count X successes in a binomial setting is a binomial random variable. The probability distribution of X is a binomial distribution with parameters n and p, where n is the number of trials of the chance process (experiment) and p is the probability of success on any one trial. The possible values of X are from 0 to n. Ex. #2 In which of the following situations, does the random variable have a binomial distribution? If the variable does not have a binomial distribution, indicate why. a) Shuffle a deck of cards. Turn over the top card. Put the card back in the deck, and shuffle again. Repeat this process 10 times. Let X= the number of aces that you observe. b) Choose three students at random from your class. Let Y= be the number who are over 6 feet tall. c) Flip a coin. If it s heads, roll a 6-sided die. If it s tails, roll and 8=sided die. Repeat this process 5 times. Let W= the number of 5 s you roll. d) Genetics says that children receive genes from each of their parents independently. Each child of a particular pair of parents has a probability of 0.25 of having type O blood. Suppose these parents have 5 children. Let X= the number of children with type O blood.
Ex. #3 Consider tossing a biased coin 4 times and recording the number of heads. a) Create a list of all possible outcomes of this experiment. b) Define X to be the number of heads you observe in the 4 tosses. Given P(head) = 0.6. find the following: P(X=0) P(X=1) P(X=2) P(X=3) P(X=4) c) Is the probability distribution in part b) a legitimate probability distribution? Why? The binomial coefficient: The number of ways of arranging k successes among n observations is given by the binomial coefficient: n n! = nc = k k k!( n k)! ** note: you only need to be able to calculate this using your calculator ( n MATH->NUM n C r r) Binomial Probability: If X has the binomial distribution with n trials and the probabilyt p of success on each trial, the possible values of X are 0, 1, 2,, n. If k is any one of these values:
Mean and Standard Deviation of a Binomial Distribution: Ex. #4 A student writes a 5 question multiple choice test by guessing on each answer. Each question has 4 possible answers. a) Use the binomial probability model to find the distribution to find the probability distribution of X, where X is the number of correct answers on the test. X 0 1 2 3 4 5 P(X) b) Make a histogram of the probability distribution. Describe what you see. c) What would you expect to be the mean of X? d) Calculate the mean and standard deviation for X.
Using the Binomial Distribution for Statistical Sampling Ex. #5 An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches. Suppose that 10% of the switches in the shipment are bad, however the engineer is not aware of this statistic. The engineer decides to estimate the percentage of bad switches in the shipment by counting the number of bad switches in his sample. a) Is this a binomial setting? b) What is the probability of getting no defectives in the sample? Sampling without replacement condition: When taking a SRS of size n from a population of size N, we can use the binomial distribution to model the count of successes in the sample as long as: Ex. #6 An airline has just finished training 25 first officers - 15 male and 10 female to become captains. Unfortunately, only eight captain positions are available right now. Airline managers decide to use a lottery to determine which pilots will fill the available positions. Of the 8 captains chosen, 5 are female and 3 are male. Explain why the probability that 5 female pilots are chosen in a fair lottery is NOT: 8 P( x = 5) = (.40) (.60) 5 5 3 Note: we will not cover the Normal approximation to the Binomial Distribution at this time (pg. 395-397) Homework: Pg. 403 #69, 71, 73, 75, 77, 79, 81, 83, 85, 87