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6.2 Binomial Probability Distribution

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Objectives:

By the end of this section, I will be

able to…

- Explain what constitutes a binomial experiment.
- Compute probabilities using the binomial probability formula.
- Find probabilities using the binomial tables.
- Calculate and interpret the mean, variance, and standard deviation of the binomial random variable.

- There 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

Only two outcomes possible

Fixed number of trials

Outcomes are independent of one another

Probability of a success remains the same for each trial

Binomial Dist.

- You are going to toss a coin 4 times. Determine how many times you get tails.
- Sallies is in the bonus for a 1 and 1 situation. Michael is an 82% shooter. Michael’s outcome is noted.
- 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.

Ask yourself the four questions.

Are there 2 outcomes?

NOT a Binomial Dist.

Is there a fixed number of trials?

Are the outcomes independent?

Does the probability remain the same for any trial?

Binomial Dist.

p(success) = p

p(failure) = q

n = total number of trials

x = selected number out of total

- What is the most difficult self-discipline challenge for people?
- The human brain is 80% of this.
- Bluebirds cannot see this color.
- Flamingos do this to cool down.
- Who were the first father son pair to be elected President of the USA?

- What is the most difficult self-discipline challenge for people? WEIGHT
- The human brain is 80% of WATER
- Bluebirds cannot see this color. BLUE
- Flamingos do this to themselves to cool down.? PEE on themselves
- 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?

1) What is the probability that exactlyseven of you passed the quiz?

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 “exactly4.”

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

μ

= mean

n = total number

p = probability of success

- The variance of a binomial distribution is:
- The standard deviation of a binomial distribution is:

1) A die is rolled 600 times. Find the mean, variance, and standard deviation of the number of 4’s that are rolled.

n = total number = 600

p = p(success) = p(4) = 1/6

q = p(failure) = p(not 4) = 5/6

mean = 600(1/6) = 100

variance = 600(1/6)(5/6) ≈ 83.333

- 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)?
- Find the variance and standard deviaton.

n = total number = 500

p = p(success) = p(cancel) = 7%

q = p(failure) = p(not cancel) = 93%

mean = 500(7%) = 35

variance = 500(7%)(93%) = 32.55