Section 6.3Section 6.3

Binomial DistributionsBinomial Distributions

A Gaggle of GirlsA Gaggle of Girls

Let’s use simulation to find the probability Let’s use simulation to find the probability that a couple who has three children has that a couple who has three children has all girls.all girls.P(girl) = 0.5P(girl) = 0.5Let 0 = boy and 1 = girl.Let 0 = boy and 1 = girl.Use your calculator to choose 3 random Use your calculator to choose 3 random digits to simulate this experiment.digits to simulate this experiment.Complete this experiment 50 times in your Complete this experiment 50 times in your group and record. Create a probability group and record. Create a probability distribution for X = number of girls.distribution for X = number of girls.

Gaggle continuedGaggle continued

What was your group’s probability for What was your group’s probability for having three girls?having three girls?

Use your knowledge of probabilities to find Use your knowledge of probabilities to find the actual chance that a family with three the actual chance that a family with three children has three girls.children has three girls.

Are these close?Are these close?

Children, Again???Children, Again???

Two types of scenarios:Two types of scenarios: A couple is going to have children until they A couple is going to have children until they

have a girl.have a girl.Here, the random variable is how many children Here, the random variable is how many children will it take to get a girl.will it take to get a girl.

A couple is going to have 3 children and we’ll A couple is going to have 3 children and we’ll count how many are girls.count how many are girls.

Here, the random variable is how many girls there Here, the random variable is how many girls there are out of the 3 children.are out of the 3 children.

Dichotomous OutcomesDichotomous Outcomes

Both of those situations have Both of those situations have dichotomous dichotomous (two) outcomes. (two) outcomes.

Other examples with two outcomes:Other examples with two outcomes: Coin toss (heads or tails)Coin toss (heads or tails) Shooting free throws (make or miss)Shooting free throws (make or miss) A game of baseball (win or lose)A game of baseball (win or lose)

Special Type of SettingSpecial Type of Setting

In this chapter, we’ll study a setting with In this chapter, we’ll study a setting with two outcomes where there are a fixed two outcomes where there are a fixed number of observations (or trials).number of observations (or trials).

The The binomial distributionbinomial distribution is a special is a special type of setting in which there are two type of setting in which there are two outcomes of interest. outcomes of interest.

4 Conditions for a Binomial Setting4 Conditions for a Binomial Setting

1.1. There are two outcomes for each There are two outcomes for each observation, which we call “success” or observation, which we call “success” or “failure.”“failure.”

2.2. There is a fixed number There is a fixed number nn of of observations.observations.

3.3. The The nn observations are all independent. observations are all independent.

4.4. The probability of success, called The probability of success, called pp, is , is the same for each observation.the same for each observation.

Binomial Random VariablesBinomial Random Variables

Binomial random variableBinomial random variable: In a binomial setting, : In a binomial setting, the random variable X = # of success. the random variable X = # of success. The probability distribution of X is called a The probability distribution of X is called a binomial distributionbinomial distribution.. The parameters of a binomial distribution are The parameters of a binomial distribution are nn (the (the

number of observations) and number of observations) and pp (the probability of (the probability of success on any one observation).success on any one observation).

B(n, p)B(n, p)Is a binomial random variable discrete or Is a binomial random variable discrete or continuous?continuous?

Discrete…

ExampleExample

Blood type is inherited. If both parents Blood type is inherited. If both parents have the genes for the O and A blood have the genes for the O and A blood types, then each child has probability 0.25 types, then each child has probability 0.25 of getting two O genes and thus having of getting two O genes and thus having type O blood. Is the number of O blood type O blood. Is the number of O blood types among this couple’s 5 children a types among this couple’s 5 children a binomial distribution?binomial distribution? If so, what are If so, what are nn and and pp?? If not, why not?If not, why not?

ExampleExample

Deal 10 cards from a well-shuffled deck of Deal 10 cards from a well-shuffled deck of cards. Let X = the number of red cards. Is cards. Let X = the number of red cards. Is this a binomial distribution?this a binomial distribution? If so, what are If so, what are nn and and pp?? If not, why not?If not, why not?

Using the Calculator to Find Using the Calculator to Find Binomial ProbabilitiesBinomial Probabilities

Under 2Under 2ndnd VARS (DISTR), find 0:binompdf( VARS (DISTR), find 0:binompdf(

This command finds probabilities for the This command finds probabilities for the binomial binomial pprobability robability ddistribution istribution ffunction.unction.

The parameters for this command are The parameters for this command are binomialpdf(n, p, x) IN THAT ORDER.binomialpdf(n, p, x) IN THAT ORDER. This will only give you the probability of a This will only give you the probability of a

single x value.single x value.

ExampleExample

Let’s go back to the couple having three Let’s go back to the couple having three children. Let X = the number of girls.children. Let X = the number of girls.

p = P(success) = P(girl) = 0.5p = P(success) = P(girl) = 0.5

The possible values for X is 0, 1, 2, 3.The possible values for X is 0, 1, 2, 3.

Using the binompdf(n,p,x) command, Using the binompdf(n,p,x) command, complete the probability distribution.complete the probability distribution.

What is the probability that the couple will What is the probability that the couple will have no more than 1 girl?have no more than 1 girl?

Cumulative Distribution FunctionCumulative Distribution Function

The pdf command lets you find The pdf command lets you find probabilities for ONE value of X at a time.probabilities for ONE value of X at a time.binomialcdf(n, p, x)binomialcdf(n, p, x) This time, you will be given the sum of the This time, you will be given the sum of the

probabilities ≤ x. Be sure you remember this probabilities ≤ x. Be sure you remember this when answering a questionwhen answering a question

The cdf command finds cumulative The cdf command finds cumulative probabilities. We can use it to quickly find probabilities. We can use it to quickly find probabilities such as P(X < 7) or P(X ≥ 4).probabilities such as P(X < 7) or P(X ≥ 4).

Corinne’s Free ThrowsCorinne’s Free Throws

Corinne makes 75% of her free throws Corinne makes 75% of her free throws over the course of a season. In a key over the course of a season. In a key game, she shoots 12 free throws and game, she shoots 12 free throws and makes 7 of them. Is it unusual for her to makes 7 of them. Is it unusual for her to shoot this poorly or worse?shoot this poorly or worse?

What is the probability that Corinne makes What is the probability that Corinne makes at least 6 of the 12 free throws?at least 6 of the 12 free throws?

HomeworkHomework

Chapter 6#Chapter 6#

69-72, 86, 94 69-72, 86, 94