Chapter 8 Section 8.1 Binomial Distribution Target: The student will know what the 4 characteristics are of a binomial distribution and understand how to use them to identify a binomial setting. Process Skill: Predict Big Rock: Probability Essential Question: What is the probability of passing a 10 question multiple choice test that has answers A, B, C, D, or E? Homework: Page 516: 8.1 8.6 Page 519: 8.7 8.12 Page 523: 8.13 8.16 Jan 17 7:08 PM Pop Quiz Results # correct 0 1 2 3 4 5 6 7 8 9 10 Tally How many times did someone "pass" the quiz? Total number of pass: Total number of simulations: Jan 17 7:10 PM Example: We use a coin toss to see which of two football teams gets the choice of kicking off or receiving to begin the game. A basketball player shoots a free throw; the outcomes of interest are {she makes the shot; she misses} A young couple prepares for their first child; the possible outcomes are {boy; girl}. Jan 17 7:17 PM 1
Example: Binomial? Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability of.25 of getting two O genes and so having type O blood. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X of successes in 5 independent observations with probability.25 of a success on each observation. X has the binomial distribution with n = 5 and p = 0.25. Or B(5, 0.25) Jan 17 7:21 PM Binomial? Deal 10 cards from a shuffled deck and count the number X of red cards. There are 10 observations, and each gives either a red or a black card. A "success" is a red card. Is this binomial? Jan 17 7:25 PM Binomial Settings in Statistical sampling: Inspecting switches An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts the number of X bad switches in the sample. Is this a binomial setting? B(10,.1) Jan 17 7:27 PM 2
Aircraft engine reliability Binomial setting? Engineers define reliability as the probability that an item will perform its function under specific conditions for a specific period of time. If an aircraft engine turbine has probability 0.999 of performing properly for an hour of flight, the number of turbines in a fleet of 350 engines that fly for an hour without failure has the B(350, 0.999) distribution. This is only true if we can assume independence. Jan 17 7:29 PM Back to blood type: Each child born to a particular set of parents has probability 0.25 of having type O blood. If these parents have 5 children, what is the probability that exactly 2 of them have type O blood? B(5, 0.25) and we want P(X = 2) Use S for success and F for failure. Step 1: Find the probability that a specific 2 of the 5 tries give successes, say the first and the third. (0.25)(0.75)(0.25)(0.75)(0.75) or (0.25) 2 (0.75) 3 Jan 17 7:33 PM Step 2: It does not matter what two are the successes, the probability will always be the same, for example if it were the last two: (0.75)(0.75)(0.75)(0.25)(0.25) = 0.25 2 * 0.25 3 But the question is, how many different ways can we get exactly two of the children with a success? SSFFF SFSFF SFFSF SFFFS FSSFF FSFSF FSFFS FFSSF FFSFS FFFSS There are 10 combination, therefore P(X = 2) = 10(0.25)^2(0.75)^3 = 0.2637 Jan 17 7:38 PM 3
Jan 17 7:43 PM We will ignore the formula and go straight to the calculator for this. binompdf( Total number of objects we are dealing with, probability of success, how many successes we are interested in) The binompdf gives you only the exact probability. Example: For the 5 children, we wanted to know the success of getting exactly 2 children with type O blood. Binompdf(5, 0.25, 2) Jan 17 7:44 PM Inspecting Switches A quality engineer selects an SRS of 10 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet specifications. What is the probability that no more than 1 of the 10 switches in the sample fail inspections? B(10, 0.1) is the distribution. We want P(X less than or equal to 1) = P(x = 0) + P(x = 1) binompdf(10, 0.1, 0) + binompdf(10, 0.1, 1) Jan 17 7:49 PM 4
When we add the probability of an event and everything before it, we call this a Cumulative distribution function, or a cdf. Jan 17 7:53 PM Corinne's free throws Corinne is a basketball player who makes 75% of her free throws over the course of the season. In a key game, corinne shoots 12 free throws and makes only 7 of them. The fans think that she failed because she was nervous. Is it unusual for Corinne to perform this poorly? What is P(X is less than or equal to 7)? = P(X = 0) + P(X = 1) + P( X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) B(12, 0.75) There is an easier way to do this. binomcdf(12, 0.75, 7) Jan 17 7:55 PM PDF VS CDF Lets look at the pdf table and compare this to the cdf table for Corinne. PDF X: 0 1 2 3 4 5 6 7 8 9 10 11 12 P(X) 0.0 0.0 0.0 0.0.002.011.040 0.103 0.194 0.258 0.232 0.127 0.032 CDF X: 0 1 2 3 4 5 6 7 8 9 10 11 12 P(X) 0.0 0.0 0.0 0.0.003.014.054 0.158 0.351 0.609 0.842 0.968 1 Jan 17 8:01 PM 5
AP Statistics Chapter 8 Section 8.1 Mean and Standard Deviation of a binomial distribution Target: The student will know how to find the mean and standard deviation of a binomial distribution and understand how to interpret these in context of the problem given. Process Skill: Describe Big Rock: Probability Essential Question: What is the expected number of free throws for Kiz to make in a basketball game? Jan 17 8:25 PM Example: Bad switches Continuing our example of bad switches from yesterday, B(10, 0.1) What are the mean and standard deviation? Jan 17 8:27 PM Approximating the Normal Curve with a Binomial Distribution The formula for the binomial distribution becomes very long and tedious when we have a large number of observations. However, as our sample size gets very large, the binomial distribution starts to look like the Normal Curve. Example: Attitudes toward shopping Are attitudes toward shopping changing? Sample surveys show that fewer people enjoy shopping than in the past. A survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that "I like to buy new clothes but shopping is often frustrating and timeconsuming." The population that the poll want to draw conclusions about is all U.S. residents aged 18 and over. Suppose that in fact 60% of all adult U.S. residents would say "Agree" if asked the same question. What is the probability that 1520 or more of the sample agree? P(X greater than or equal to 1520) = 1 P(X less than or equal to 1519) binomcdf (2500, 0.6, 1519) = 0.7869 1 0.7869 = 0.2131 Jan 17 8:30 PM 6
We can look at this as a normal curve instead. What is the expected value or mean of this sample? What is the standard deviation? S.D. = 24.49 Mean = 1500 N(1500, 24.49) Jan 17 8:38 PM Homework: Page 529: 8.19 8.24 Jan 17 8:41 PM 7