Binomial probability model describes the number of successes in a specified number of trials.

1. The Binomial Probability Model Binomial probability model describes the number of successes in a specified number of trials. You need: * 2 parameters (success, failure) * Number of trials, n * probability of success, p

2. Example: A cereal manufacturer puts cards of 3 famous athletes in its cereal boxes. 20% of the boxes contain pictures of Derek Jeter, 30% contain a picture of David Beckham, and the rest contain a picture of Serena Williams. Suppose you want to know what the probability is of getting 2 Derek Jeter cards if you buy 5 boxes. Could you use a binomial probability model?

3. We can use a binomial probability model because: 1 – there are two outcomes (success – getting a Derek Jeter card, failure – not getting a Derek Jeter card. 2 – there is a set number of trials (n = 5) 3 – we know the probability of success (p = .2)

4. Using the model: Let X = the random variable Let n = 5 Let p = .2 (the probability of 1 success) Let k = 2 (the number of successes) 2 successes in 5 trials means 2 successes & 3 failures. The possible order in which the outcomes can occur are disjoint (e.g., if 2 successes came in the first 2 trials, they couldn’t come in the last 2).

5. Each different order in which we can have k successes in n trials is called a combination. It can be represented by nCk (n choose k) or . To figure out the number of combinations in our trial: This means there are 10 ways to get 2 Derek Jeter pictures in 5 boxes.

6. Next step Probability of success k Probability of failure (n- k) Nbr of successes Nbr of combinations The probability of success in getting 2 Derek Jeter cards in 5 boxes of cereal is .2048.

7. Another example: The count of X children with type O blood among 5 children whose parents carry genes for both the O and the A blood types is B(5, 0.25). Find P(X=3). Note: B means binomial setting, n = 5, p = 0.25

8. Note: 0! = 1

9. One step further: Suppose the number of electrical switches that fail inspection is B(10, .1). What is the probability that no more than 1 switch fails.

10. = np µ σ These formulas are only good for binomial distributions. They can’t be used for other discrete random variables.

11. Example: Using the previous cereal example, in 100 boxes of cereal, how many Derek Jeter cards do you expect to find? Step 1: E(x)= np = (100)(.2) = 20 Step 2: Step 3: Summary: In 100 cereal boxes, we expect to find 20 Derek Jeter cards, with a standard deviation of 4 cards.

12. Rule of thumb: If np ≥ 10 and n(1-p) ≥ 10, Normal approximation can be used. Ex: (from your book page 527) A survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying clothes but shopping is often frustrating and time consuming.” Suppose that in fact 60% of all adult U.S. residents would say “Agree” if asked the same question. What is the probabiltiy that 1520 or more would agree?

13. Since both criteria for Normal Approximation are satisfied, we can use Normal distribution calculations. µ = (2500)(.6) = 1500 σ = = 24.49

14. Using the calculator 1) Probability distribution function – given a discrete random variable X, the probability distribution function assigns a probability to each value of X. See page 520, example 8.7 for binompdf(n,p,x) 2) Cumulative distribution function – given a random variable X, the cumulative distribution function of X calculates the sum of probability of obtaining at most X successes in n trials. See page 522 example 8.10 for binomcdf(n,p,x) 3) Look at the Technology Tool Box pages 530 - 532