Introduction to Probability and Statistics Chapter 5 Discrete

Introduction to Probability and Statistics Chapter 5 Discrete Distributions

Discrete Random Variables • Discrete random variables take on only a finite or countable many of values. üNumber of heads in 1000 trials of coin tossing üNumber of cars that enter UNI in a certain day

Binomial Random Variable • The coin-tossing experiment is a simple example of a binomial random variable. Toss a fair coin n = 3 times and record • x = number of heads. x 0 1 2 3 p(x) 1/8 3/8 1/8

Example • Toss a coin 10 times • For each single trial, probability of getting a head is 0. 4 • Let x denote the number of heads

The Binomial Experiment 1. The experiment consists of n identical trials. 2. Each trial results in one of two outcomes, outcomes success (S) or failure (F). 3. Probability of success on a single trial is p and remains constant from trial to trial. The probability of failure is q = 1 – p. 4. Trials are independent 5. Random variable x, the number of successes in n trials. x – Binomial random variable with parameters n and p

Binomial or Not? m m mm • A box contains 4 green M&Ms and 5 red ones • Take out 3 with replacement • x denotes number of greens • Is x binomial? Yes, 3 trials are independent with same probability of getting a green.

Binomial or Not? m m mm • A box contains 4 green M&Ms and 5 red ones • Take out 3 without replacement • x denotes number of greens • Is x binomial? NO, when we take out the second M&M, the probability of getting a green depends on color of the first. 3 trials are dependent.

Binomial or Not? • Very few real life applications satisfy these requirements exactly. • Select 10 people from the U. S. population, and suppose that 15% of the population has the Alzheimer’s gene. • For the first person, p = P(gene) =. 15 • For the second person, p P(gene) =. 15, even though one person has been removed from the population… • For the tenth person, p P(gene) =. 15 Yes, independent trials with the same probability of success

Binomial Random Variable • Rule of Thumb: Sample size n; Population size N; If n/N <. 05, the experiment is Binomial. • Example: A geneticist samples 10 people and x counts the number who have a gene linked to Alzheimer’s disease. • Success: • Number of n = 10 Has gene trials: • Probability of • Failure: Success p = P(has gene) = 0. 15 Doesn’t have gene

Example • Toss a coin 10 times • For each single trial, probability of getting a head is 0. 4 • Let x denote the number of heads Find probability of getting exactly 3 heads. i. e. P(x=3). Find probability distribution of x

Solution • Simple events: Strings of H’s and T’s with length 10 • Event A: {strings with exactly 3 H’s}; HTTTHTHTTT TTHHTTTTHT… • Probability of getting a given string in A: HTTTHTHTTT • Number of strings in A • Probability of event A. i. e. P(x=3)

A General Example • Toss a coin n times; For each single trial, probability of getting a head is p; • Let x denote the number of heads; Find the probability of getting exactly k heads. i. e. P(x=k) Find probability distribution of x.

Binomial Probability Distribution • For a binomial experiment with n trials and probability p of success on a given trial, the probability of k successes in n trials is

Binomial Mean, Variance and Standard Deviation • For a binomial experiment with n trials and probability p of success on a given trial, the measures of center and spread are:

Example A marksman hits a target 80% of the time. He fires 5 shots at the target. What is the probability that exactly 3 shots hit the target? n= 5 success = hit p =. 8 x = # of hits

Example What is the probability that more than 3 shots hit the target?

Example • x = number of hits. • What are the mean and standard deviation for x? (n=5, p=. 8) m

Cumulative Probability You can use the cumulative probability tables to find probabilities for selected binomial distributions. üBinomial cumulative probability: P(x k) = P(x = 0) +…+ P(x = k)

Key Concepts I. The Binomial Random Variable 1. Five characteristics: the experiment consists of n identical trials; each resulting in either success S or failure F; probability of success is p and remains constant; all trials are independent; x is the number of successes in n trials. 2. Calculating binomial probabilities a. Formula: b. Cumulative binomial probability P(x k). 3. Mean of the binomial random variable: 4. Variance and standard deviation:

Example According to the Humane Society of the United States, there approximately 40% of U. S. households own dogs. Suppose 15 households are selected at random. Find 1. 2. 3. 4. probability that exactly 8 households own dogs? probability that at most 3 households own dogs? probability that more than 10 own dogs? the mean, variance and standard deviation of x, the number of households that own dogs.

Example According to the Humane Society of the United States, there approximately 40% of U. S. households own dogs. Suppose 15 households are selected at random. What is probability that exactly 8 households own dogs? n= 15 success = own dog p =. 4 x= # households that own dog

Example What is the probability that at most 3 households own dogs?

Example What are the mean, variance and standard deviation of random variable x? (n=15, p=. 4)

Binomial Probability • Probability distribution for Binomial random variable x with n=15, p=0. 4

1. Example What are the mean, variance and standard deviation of random variable x? 2. Calculate interval within 2 standard deviations of mean. What values fall into this interval? 3. Find the probability that x fall into this interval.