Warm-upGrab a die and roll it 10 times

and record how many times you roll a 5. Repeat this 7 times and record results.

This time roll the die until you get a 5. Record how many rolls it took you. Repeat this 7 times and record results.

Warm-UpWhat’s the smallest number you

can get for the first one?What’s the largest number?

What’s the smallest number you can get for the second one?

What’s the largest number?

6.2 Homework Questions

Section 6.3Binomial Random Variables

What does “bi” mean?List all of the words you can think

of that start with “bi”…

Binomial SettingThe four conditions for a binomial setting

are:

1. Success/Failure2. Independent Trials3. Constant “p” (probability of success)4. Set number of trials, n

GeometricThe four conditions for a

geometric setting are:

1. Success/Failure2. Independent Trials3. Constant “p” (probability of

success)4. No set number of trials, n

Binomial Random VariableThe count X of successes in a

binomial setting is a binomial random variable. The probability distribution of X is a binomial distribution with parameters n and p. The possible values of X are the whole numbers from 0 to n.

Binomial? Genetics says that children receive genes from

each of their parents independently. Each child of a particular pair of parents has probability 0.25 of having type O blood. Suppose these parents have 5 children. Let X = the number of children with type O blood.

Shuffle a deck of cards. Turn over the first 10 cards, one at a time. Let Y = the number of aces you observe.

Shuffle a deck of cards. Turn over the top card. Put the card back in the deck, and shuffle again. Repeat this process until you get an ace. Let W = the number of cards required.

Binomial ProbabilitiesLet’s do the children’s gene

problem…P(none of the children have type

O)=

P(x=1)

Building the formulaP(x = k) = P(exactly k successes

in a trial)= number of arrangements

Number of arrangements:Binomial CoefficientThe number of ways of arranging k

successes among n observations is given by the binomial coefficient:

Do you remember what n! means?CAUTION - has nothing to do with

For example…When x = 1, we had 5

arrangements…

There is a button on your calculator!

5 nCr 1Math – Prob – nCr

Binomial ProbabilityIf X has the binomial distribution

with n trials and probability p of success on each trial, the possible values of X are 0, 1, 2,…, n. If k is any one of these values,

This is on the formula sheet!

ExampleFind the probability that exactly 3

children have type O blood.

Should the parents be surprised if more than 3 of their children have type O blood? Justify your answer.

Mean and Standard Deviation of a Binomial DistributionBlood Type Probability

Distribution:X 0 1 2 3 4 5

P(X) 0.23730

0.39551

0.26367

0.08789

0.01465

0.00098

Mean and Standard Deviation of Binomial Random VariablesIf a count X has the binomial

distribution with number of trials n and probability of success p, the mean and standard deviation of X are:

Remember – these formulas ONLY work for binomial distributions!

Homework #3Together, let’s do numbers 69-72Pg. 403 (73-75, 77, 79, 80, 82, 84-87,

89-92, 94-105)

Warm-Up

Normal Approximation for Binomial DistributionsAs a rule of thumb, we will use

the Normal approximation when n is so large that:

That is, the expected number of successes and failures are both at least 10.

ExampleSuppose that exactly 60% of all

adult US residents would say “agree” if asked if they think shopping is frustrating. A survey asked nationwide sampled 2500 adults.

Let X = the number of people who agree.◦Show that X is approximately a

binomial random variable.

ExampleCheck the conditions for using a

Normal approximation in this setting.

ExampleUse a Normal distribution to

estimate the probability that 1520 or more of the sample agree.

Homework #3Together, let’s do numbers 69-72Pg. 403 (73-75, 77, 79, 80, 82, 84-87,

89-92, 94-105)