Math Tech IIII, May 4 The Binomial Distribution IV Book Sections: 4.2 Essential Questions: How can I compute the probability of any event? What do I need to know about the binomial distribution to pass the Unit 7 Test? Standards: DA-5.6, S.MD.1,.2,.3

The List of Knows Know the following: Know the four binomial conditions/characteristics Recognize a binomial distribution, or when it applies in a probability problem Be able to compute binomial probability Know what is meant by a cumulative binomial distribution and when it applies Be able to create and use a binomial probability distribution Know how to compute the mean (μ) and standard deviation (σ) of a binomial distribution

The Universal First Step Identify n, p, and x (if it applies) or all possible values of x in your problem. p may be given or it may not. If not, enough information will be given to figure it out. Either way, you must have p. Important point – no one is ever going to give you q. If you need it, YOU are going to have to find it. How? q = 1 - p

Any Binomial Computation The probability of any equality/inequality of x successes in n trials. Exactly x (x = ) binomialpdf(n, p, x) At most x (x ≤ ) binomialcdf(n, p, x) Use these adjustments for any other inequality binomial computation Fewer than x (x <) binomialcdf(n, p, x -1) At least x (x ≥) 1 – binomialcdf(n, p, x- 1) More than x (x >) 1 – binomialcdf(n, p, x)

Binomial Statistics Because of the nature of this distribution, binomial mean, variance, and standard deviation are almost trivial. Here are the formulas: μ = np σ 2 = npq σ = Mean Variance Standard deviation

Example 1 The mailing list of an agency that markets scuba-diving trips to Hawaii contains 65% males and 35% females. The agency calls 6 people chosen at random from their list. What is the probability that they call A)Fewer than 3 females B)More than 2 males

Example 1a The mailing list of an agency that markets scuba-diving trips to Hawaii contains 65% males and 35% females. The agency calls 10 people chosen at random from their list. If 10 people were called, what is the mean number of females who would be called: What is the standard deviation?

What Makes a Binomial Experiment? A binomial experiment is a probability experiment that satisfies the following conditions: 1.Contains a fixed number of trials that are all independent. 2.All outcomes are categorized as successes or failures. 3.The probability of a success (p) is the same for each trial. 4.There is a computation for the probability of a specific number of successes.

Binomial Notation Binomial computations are known as probability by formula. The formula has a set of arguments that you must know and understand in application. Here is that notation: SymbolDescription n The number of times a trial is repeated p The probability of success in a single trial q The probability of failure in a single trial (q = 1 – p) x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, …, n

Binomial Computation III Creating a binomial distribution and graph: To construct a binomial distribution table, open STAT Editor 1)type in 0 to n in L1 2)Move cursor to top of L2 column (so L2 is hilighted) 3)Type in command binomialpdf(n, p, L1) and L2 gets the probabilities. 4)Go to stat plot and set up appropriate graph.

Example 2 Three in five beagle puppies have their eyes open within 7 days of their birth. James’ beagle had a litter of 5 pups 7 days ago. Produce a discrete probability distribution for this binomial situation. What is the probability that 2 or 3 pups have their eyes open in 7 days?

Final Thought Probability is always a number between 0 and 1, it can be 0 and it can be 1.

Classwork: CW 5/4/15, 1-11 (Pre- Test) Homework – None