Math Tech IIII, May 4 The Binomial Distribution IV Book Sections: 4.2 Essential Questions: How can I...

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/characteristicsRecognize a binomial distribution, or when it applies in a

probability problemBe able to compute binomial probabilityKnow what is meant by a cumulative binomial distribution

and when it appliesBe able to create and use a binomial probability distributionKnow 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

σ2 = npq

σ = npq

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:

Symbol Descriptionn The number of times a trial is repeatedp The probability of success in a single trialq 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