1. AP STATISTICS LESSON 8 – 1 ( DAY 2 ) THE BINOMIAL DISTRIBUTION (BINOMIAL FORMULAS)

2. Binomial Formulas Example 8.9 page 446 Each child born to a particular set of parents has probability 0.25 of having blood type O. If these parents have 5 children, what is the probability that exactly 2 of them have type O blood?

3. ESSENTIAL QUESTION: What are the binomial formulas and how are they used to solve problems that can be modeled in binomial settings? Objectives: To define and use the binomial formulas. To derive and use binomial means and standard deviation

4. Binomial Coefficient = The number of ways of arranging k successes among n observations is given by the binomial coefficient for k = 0,1,2…..n. nknk = n! k!(n – k )!

5. Vocabulary Factorial - ! – The formula for binomial coefficients uses the factorial notation. For any positive whole number n, its factorial n! is n! = n x (n-1) x ( n – 2 ) x …x 3 x 2 x1 The notation is not related to the Fraction. A helpful way to remember its meaning is to read it as “ binomial coefficient “ n choose k.” nknk nknk

6. Binomial Probability If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, 3, …n. If k is any one of these values, P(X = k ) = p k (1 – p) n - k nknk

7. Example 8.10 Defective Switches Page 448 The number X of switches that fail inspection in Example 8.3 has approximately the binomial distribution with n = 10 and p = 0.1. Find the probability that no more than 1 fails.

8. Mean and Standard Deviation of a Binomial Random Variable If a count X has the binomial distribution with number of observations n and probability p, the mean and the standard deviation of X are μ = np σ = √ np(1 – p ) These short formulas are good only for binomial distributions. They can’t be used for other discrete random variables.

9. The Normal Approximation to Binomial Distributions The formula for binomial probabilities becomes awkward as the number of trials n increases. As the number of trials n gets larger, the binomial distribution gets close to normal distribution. When n is large, we can use normal probability calculations to approximate binomial probabilities.

10. Example 8.12 Attitudes Toward Shopping Page 452 A recent survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying new clothes, but shopping is often frustrating and time-consuming.” p = 60% and X ≤ 1520 Are attitudes toward shopping changing?

11. Normal Approximation for Binomial Distributions Suppose that a count X has the binomial distribution with n trials and success probability p. When n is large, the distribution of X ix approximately normal, N ( np,√np(1 – p) ). As a rule of thumb, we will use normal approximation when n and p satisfy np ≥ 10 and n(1-p) ≥ 10.