Binomial Distributions

Binomial Distributions Calculating the Probability of Success

Contents • How to identify binomial distributions. • How to calculate binomial probabilities. • When to use Normal approximations for binomial distributions.

1. How to identify binomial distributions Identification

Binomial Distribution • Discrete random variable • Define X • S={0, 1, 2, …} • Binomial setting • XB(n, p) • Key idea: Count success!

The Binomial Setting • “Success” or “Failure.” • Probability of success same for each trial. • Trials independent. • Fixed number of trials.

Characteristics • XB(n, p) • Expected Value: • Variance:

2. How to Calculate Binomial Probabilities Calculations

Probability Calculations Where: k is the desired count, n is the fixed number of trials, p is the probability of success, and (1-p) is the probability of failure.

Example What is the probability of tossing a fair coin five times and getting exactly three heads?

Check for Binomial Setting • Success is flipping a head;failure is flipping a tail. • The probability of flipping heads on a fair coin is 50% each time. • Each flip is independent. • There is a fixed number of trials.

Define Values In our example: k = 3 n = 5 p = 0.5 & (1-p) = 0.5

Calculations

More Calculations

Interpretation There is about a 31% chance of flipping a fair coin 5 times and getting exactly 3 heads.

Binomial Distribution Using similar calculations,we can find each probability:

3. When to use Normal approximations. Normal Approximations

Normal Approximations If n is large enough, XB(n, p)  XN(,). Follow two “rules of thumb:” • np  10, & • N(1-p)  10

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Binomial Distributions