AP Statistics Chapter 8 Section 1

A gaggle of girls The Ferrell family have 3 children: Jennifer, Jessica, and Jaclyn. If we assume that a couple is equally likely to have a girl or a boy, then how unusual is it for a family to have 3 children who are all girls? If success = girl (of course) and failure = boy (sorry), then p(success) =.5 Define random variable X as the number of girls Simulate families with 3 children to determine the long-term relative frequency of a family with 3 girls, P(X=3)

1.Let even digits represent girls: 0,2,4,6,8 Let odd digits represent boys: 1,3,5,7,9 2.RandInt (0,9,3) If all three digits are even it represents a family with three girls. Any other combination of digits represents a family with a mixture of genders or all boys. Perform 40 simulations. Combine with classmates. Calculate the relative frequency of the event {3 girls}. 3 girls Not 3 girls

X P(X) Determine the sample space for this simulation. What are the possible outcomes? X = number of girls bbg bgg bgb gbg bbb gbb ggb ggg 0 girls1 girl2 girls3 girls

Vocab Binomial Setting Binomial random variable Binomial distribution – B(n,p) pdf cdf factorial Binomial coefficient Binomial probability Mean of a binomial random variable Standard deviation of a binomial random variable

Binomial Setting 1.two categories: Success, Failure 2.fixed number of observations, n 3.all observations are independent 4.probability of success (p) is the same for all observations

Binomial? Blood Type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. These parents have 5 children. B(5,.25)

Binomial? Deal cards from a shuffled deck and count the number X of red cards. There are 10 observations, and each gives either a red or black card. Not a binomial distribution – not independent events

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 the specifications. What is the probability that no more than 1 of the 10 switches in the sample fail inspection? Binomial? 1.Pass/Fail 2.N = 10 3.Independent events? Yes, each switch inspected will succeed or fail independently of the other switches chosen. 4.P =.1 for all switches Yes, this is a binomial distribution. B(10, 0.1)

Calculator 2 nd vars Binomial pdf (10,.1) sto L2 In L1 put This means that in this inspection of 10 switches the probability that 0 of the switches fails inspection is approx. 35%. The probability that one of the switches fails is approx. 38%, etc.

So… What is the probability that no more than 1 of the switches fails inspection? So, in this inspection the probability that no more than 1 of the switches fails inspection is approx. 74%.

Histogram Turn axes off – format – axes off 2 nd y = Set viewing window

Outcomes larger than 6 do not have probability exactly 0 but their probabilities are so small that the rounded values are You can use the 1-Var stats to verify that the sum of the probabilities =1.

Corinne is a basketball player who makes 75% of her free throws over the course of a 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? Binomial? 1.miss/make 2.N =12 3.The outcome of each shot is considered independent of the success/failure of the other shots. 4.P =.75 for all shots B(12,.75)

Was the outcome determined by her nervousness? The probability of Corinne making at most 7 free throw shots out of 12 is approx. 16%. This means that approx. 1 out of every 6 games Corinne could be expected to make 7 or less free throw shots out of a possible 12.

pdf and cdf tables X pdf cdf

Binomial Formulas

Simulation Simulate 12 free throws with 75% success Let 0 represent missed and 1 represent made randBin(1,.75, 12) Out of the 12 shots in the simulation 10 were made – approx. 83%

Binomial mean and standard deviation

Find the mean & s.d. for the free throw situation