1. 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

2. Binomial Probability Distribution
Example: Flip 100 coins and count the number of heads:
There are 100 trials (100 flips of a coin)
The outcome of one flip doesn’t affect the next flip
Each trial has two outcomes – heads or tails
The probability of heads is always ½

3. 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

4. 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 = 1 - ¼ = 0.75
We’re looking for P(8)

5. The Formula

6. 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 = x = p = ¼ = q = 0.75
How would we find the P(at least 8 right)?
P(at least 8) = P(8) + P(9) + P(10) =

7. 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?

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