1. The Binomial Distribution
MDM 4U Lesson 5.4

2. RECALL:
General term of a binomial expansion:
Expected value formula

3. Definitions
Bernoulli Trial: an independent trial that has two possible outcomes, success or failure.
Binomial probability distribution:
a distribution with independent trials whose outcome is either success or failure.
The random variable is the number of successes in a given number of trials.

4. Binomial Distribution
The trials need to be identical (n trials) and indedpendent
The two possible outcomes are success or
failure. The probability for success is p, and the probability of a failure is q = 1-p.

5. General Formula
For n independant trials of an event A whose probability of success is p, the probability of exactly k successes is given by

6. Ex. #1: For the term below identify the following:
The number of trials
The probability of p success
The number of successes

7. Ex. #2: Two dice are rolled five times
Ex. #2: Two dice are rolled five times. What is the probability that doubles occur twice?

8. Expected Value of a Binomial Experiment
The expected value of a binomial experiment that consists of n Bernoulli trials with a probability of success p, on each trial is:
E(X) = np

9. Ex #3: a) A baseball player has a batting average of 0. 320
Ex #3: a) A baseball player has a batting average of Determine the probability that the baseball player does not get a hit during a game in which they had 3 at bats.

10. Determine the probability distribution for the number of hits per game.
# hits (X) 1 2 3
P(X)

11. Calculate the batters expected number of hits per game.

12. Pascal seems normal
In the situation where the probability of success is equal to 0.5 (like flipping a coin), the binomial distribution is very close to the normal distribution

13. 20th line of Pascal's triangle
As the number of trials increases, the binomial distribution becomes smoother and more normal.

14. Practice Questions
Page 385 #1, 3, 5, 7, 8ab