The Binomial Distribution MDM 4 U Lesson 5

The Binomial Distribution MDM 4 U Lesson 5. 4

RECALL: ¢ General term of a binomial expansion: ¢ Expected value formula

Definitions ¢ ¢ Bernoulli Trial: an independent trial that has two possible outcomes, success or failure. Binomial probability distribution: l l 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.

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.

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

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

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

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

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

b. # hits (X) P(X) Determine the probability distribution for the number of hits per game. 0 1 2 3

c) Calculate the batters expected number of hits per game.

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

20 th line of Pascal's triangle 1 20 190 1140 4845 15504 38760 77520 125970 167960 184756 167960 125970 77520 38760 15504 As the number of trials increases, the binomial distribution becomes smoother and more normal.

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