Chapter 8 The Binomial and Geometric Distributions

Activity 8 – page 438 Before we simulate this event, how would we determine the probability of a couple having 3 girl children? What is the theoretical probability of the family having all 3 children be girls?

Binomial Probabilities Tossing a coin to see which football team gets the choice of kicking off or receiving the to begin the game. P(winning the coin toss)=.5 A basketball player shoots a free throw; the outcomes of interest are {makes free throw, misses free throw} P(making free throw) = ? A young couple prepares for their first child; the outcomes of interest are {boy, girl} P(boy)=.5 A manufacturer inspects items off of an assembly line; outcomes of interest are {defective, not defective}

Binomial Setting 1. Each observation falls into one of just two categories, which for convenience we call “success” and “failure”. 2. There is a fixed number of trials. 3.The n observations are all independent. That is, knowing the result of one observation tells you nothing about the other observations. 4. The probability of success, call it p, is the same for each observation. 5. We are looking usually for the probability of r successes out of the n trials.

Binomial Random Variable The distribution of the count X of successes in the binomial setting 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 0 to n. As an abbreviation, we say that X is B(n,p)

Example 8.1 Read and discuss with a neighbor.

Example 8.2 Read and discuss with a neighbor.

Example 8.3 Read and discuss with a neighbor.

Example 8.4 Read and discuss with a neighbor. Be prepared to explain to the class.

Example 8.5 Read and discuss with a neighbor. Be prepared to discuss with the class.

Binompdf & Binomcdf 2nd DISTR binompdf (n,p,r) Binompdf (10,.1,1)= .387420489. The exact probability for 1 trial out of 10 tries being successful when p=.1. Binomcdf (n,p,r) cumulative probability. Probability of up to r successes out of n trials.

Example 8.6 Read and discuss with a neighbor. Be prepared to discuss with the class. How would you do this with the calculator?

Example 8.7 Read and discuss with a neighbor. Be prepared to discuss with the class.

Example 8.8 Read and discuss with a neighbor. Be prepared to discuss with the class.

Example 8.9 Read and discuss with a neighbor. Be prepared to discuss with the class.

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, …, n. If k is any of those values,

Example 8.10 Read and discuss with a neighbor. Be prepared to discuss with the class.

Binomial Mean and Standard Deviation

The Normal approximation to Binomial Distributions As the number of trials n gets larger and larger, the binomial distribution gets closer and closer to a normal distribution.

The Normal approximation to Binomial Distributions Two criteria must be met to approximate use the normal curve as an approximation of a binomial distribution. n(p) > 10 and n(1-p) >10

Example 8.12 Read and discuss with a neighbor. Be prepared to discuss with the class.

Example 8.13 Read and discuss with a neighbor. Be prepared to discuss with the class.

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 is approximately normal, N (np, )

Technology Toolbox How can we get the calculator to draw a binomial distribution for us? Pg. 456 Do problems through number 1 – 20 and 27 – 34.

8.2 Geometric Distributions Geometric Setting 1. Each observation falls into one of just two categories, which for convenience we call “success” and “failure”. 2. The probability of success, call it p, is the same for each observation. 3. The observations are all independent. 4. The variable of interest is the number of trials required to obtain the first success.

Example 8.15 Read and be prepared to explain to the class.

Example 8.16 Read and be prepared to explain to the class.

How to Calculate Geometric Probabilities If X has a geometric distribution with probability p of success and (1 – p) of failure on each observation, the possible values of X are 1, 2, 3, … If n is any one of these values, the probability that the first success occurs on the nth trial is

Example 8.17 Read and be prepared to explain to the class.

Expected Value and Variance of Geometric Distributions If X is a geometric random variable with probability of success p on each trial, then the mean, or expected value, of the random variable, that is , the expected number of trials required to get the first success is The variance of X is

Example 8.18 Read and be prepared to explain to the class.

Example 8.19 Read and be prepared to explain to the class.

Calculator Commands geometpdf(p,r) will give you the probability of the first success happening exactly on the rth trial. geometcdf (p,r) is the cumulative probability of the 1st success happening within r trials.