Binomial Probability Distribution

A binomial probability distribution results from a procedure where:

1) There are a fixed number of trials

2) The trials are independent*

3) Each trial has only two possible outcomes

4) The probabilities are the same for all trials

*when sampling without replacement, if the sample size is less than 5% of the population size, we can treat the events as if they were independent

Binomial Probability Distribution

Example: Flip 100 coins and count the number of heads:

1. There are 100 trials (100 flips of a coin)

2. The outcome of one flip doesn’t affect the next flip

3. Each trial has two outcomes – heads or tails

4. The probability of heads is always ½

Notation

We usually call the two outcomes success and failure

p probability of success in one trial

q = 1-p probability of failure in one trial

n number of trials

x specific number of successes in n trials (can be 0 to n)

P(x) probability of exactly x successes in the n trials

Example

What’s the probability of randomly guessing 8 questions right on a 10 question multiple choice test, where each question has 4 possible answers.

 n = 10x = 8p = ¼ = 0.25q = 1 - ¼ = 0.75We’re looking for P(8)

The Formula

xnxxn qpCxP )(

Example

What’s the probability of randomly guessing 8 questions right on a 10 question multiple choice test, where each question has 4 possible answers.

n = 10 x = 8 p = ¼ = 0.25 q = 0.75

000386.075.025.0)8( 8108810 CP

How would we find the P(at least 8 right)?

P(at least 8) = P(8) + P(9) + P(10) =0.000416

You try

A company makes widgets, with a 2% defect rate. If you sample and test 15 widgets, what is the probability that:

Exactly one has a defect?

None have defects?

At least one has a defect?

Homework

4.3: 1, 5, 7, 17, 19, 21, 25, 29, 33