C HAPTER 8 Section 8.1 Part 1 – The Binomial Distribution

I NTRODUCTION In practice, we frequently encounter experimental situations where there are two outcomes of interest. Some examples are: We use a coin toss to answer a question. A basketball player shoots a free throw. A young couple prepares for their first child.

T HE B INOMIAL S ETTING 1. Each observation falls into one of just two categories, which for convenience we call “success” or “failure.” 2. There is a fixed number n of observations. 3. The n observations are all independent. (That is, knowing the results of one observation tells you nothing about the other observations). 4. The probability of success, call it p, is the same for each observation.

B INOMIAL D ISTRIBUTION The distribution of the count X of successes in the binomial settings is the binomial distribution with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that X is B ( n, p ).

E XAMPLE 8.1 -B LOOD T YPES Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability of 0.25 of getting two O genes and so of having blood type O. Suppose there are 5 children and that the children inherit independently of each other. Is this a binomial setting? If so, find n, p and X. n = 5 p =.25 X = B (5,.25)

E XAMPLE 8.2 – D EALING C ARDS redblack Deal 10 cards from a shuffled deck and count the number X of red cards. There are 10 observations and each are either a red or a black card. Is this a binomial distribution? No because each card chosen after the first is dependent on the previous pick If so what are the variables n, p and X ? None

E XAMPLE 8.3 – I NSPECTING S WITCHES 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 X of bad switches in the sample. Is this a binomial situation? Justify your answer. While each switch removed will change the proportion, it has very little effect since the shipment is so large. In this case the distribution of X is very close to the binomial distribution B (10,.1)

E XAMPLE 8.5 – I NSPECTING S WITCHES 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. What is the probability that no more than 1 of the 10 switches in the sample fail inspection? See explanation/diagram on p.442

PDF “P ROBABILITY D ISTRIBUTION F UNCTION ” Given a discrete random variable X, the probability distribution function assigns a probability to each value of X. The probability must satisfy the rules for probabilities given in Chapter 6. The TI-83 command binomPdf(n, p, X) will perform the calculations. This is found under 2 nd /DISTR/0…(or A for TI-84)

E XAMPLE 8.6 – C ORINNE ’ S F REE T HROWS

E XAMPLE 8.7 – T HREE G IRLS

The cumulative binomial probability is useful in a situation of a range of probabilities. Given a random variable X, the cumulative distribution function (cdf) of X calculates the sum of the probabilities for 0,1,2,…, up to the value X. That is, it calculates the probability of obtaining at most X successes in n trials. CDF “C UMULATIVE D ISTRIBUTION F UNCTION ”

BINOMPDF VS BINOMCDF

U SING P DF & C DF T O F IND P ROBABILITIES

Part 1 HW: P #’s 1, 2, 3, 5, & 6