6.2 Binomial Probability DistributionObjectives:By the end of this section, I will beable to…

1) Explain what constitutes a binomial experiment.

2) Compute probabilities using the binomial probability formula.

3) Find probabilities using the binomial tables.4) Calculate and interpret the mean, variance,

and standard deviation of the binomial random variable.

LifeThere are many times in life when

there are only two possible outcomes.

A baby is about to be born. Boy or Girl?

Basketball player is fouled. Make or miss?

You are about to ask a girl out. Yes or no?

BINOMIAL DISTRIBUTION

1. Only two outcomes possible2. Fixed number of trials3. Outcomes are independent of one

another4. Probability of a success remains the

same for each trial

Binomial Distribution or not?1. You are going to toss a coin 4 times.

Determine how many times you get tails.

2. Sallies is in the bonus for a 1 and 1 situation. Michael is an 82% shooter. Michael’s outcome is noted.

3. From Del State Police: 16% of victims of car accidents were minors (under 21). A random sample of 100 accidents is selected and the number of minors killed is noted.

Binomial Dist.

NOT a Binomial Dist.

Binomial Dist.

Ask yourself the four questions.

Are there 2 outcomes?Is there a fixed number of trials?Are the outcomes independent?Does the probability remain the same for any trial?

Binomial Distribution formula

p(success) = pp(failure) = qn = total number of trialsx = selected number out of

totalx n x

n xC p q

Quiz time1. What is the most difficult self-

discipline challenge for people? 2. The human brain is 80% of this.3. Bluebirds cannot see this color.4. Flamingos do this to cool down.5. Who were the first father son pair to

be elected President of the USA?

ANSWER MAN1. What is the most difficult self-discipline

challenge for people? WEIGHT

2. The human brain is 80% of WATER

3. Bluebirds cannot see this color. BLUE

4. Flamingos do this to themselves to cool

down.? PEE on themselves

5. Father Son presidents? John Adams and

John Quincy Adams

Passing Grade is 3, 4, or 5 correct.

How many students passed?

How many students failed?

EXACTLY1) What is the probability that exactly seven of you passed the quiz?

AT MOST and AT LEAST

4) What is the probability that at most three of you passed the test?

5) What is the probability that at least nineteen of you passed the test?

This is the same as saying “exactly 4.”

This is the same as saying “at most 2.”

= mean

n = total number

p = probability of success

Mean of the Binomial Distribution

x np

x μ

Variance and Standard Deviation

The variance of a binomial distribution is:

The standard deviation of a binomial distribution is:

variance npq

. . variances d npq

DICE GAMES1) A die is rolled 600 times. Find the

mean, variance, and standard

deviation of the number of 4’s that

are rolled.

x np

n = total number = 600p = p(success) = p(4) = 1/6

mean = 600(1/6) = 100

variance npqq = p(failure) = p(not 4) = 5/6

variance = 600(1/6)(5/6) ≈ 83.333

. . variances d npq 83.3333 9.1287

VACATION!!!!2) Liberty Travel finds that 7% of all vacation

packages are canceled each year. If they estimate

500 packages a year, how many can they expect

will be canceled (aka find the mean)?

3) Find the variance and standard deviaton.n = total number = 500p = p(success) = p(cancel) = 7%q = p(failure) = p(not cancel) = 93%mean = 500(7%) = 35variance = 500(7%)(93%) = 32.55

32.55 5.705