6.2 Homework Questions

Section 6.3Binomial Random Variables

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 of X 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.n=5 , p= 0.25 or B(5, 0.25)

P(X=0) P(none of the children have type O)=

P(X=1) P(one child has type O)

P(X=2)

Building the formula

P(X = k) == P(exactly k successes in “n” trials)== (number of arrangements)

So we need a nice way of finding the “number of arrangements”

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

successes among n observations is given by the binomial coefficient:

You may know this as nCr

CAUTION : is NOT the fraction

Binomial ProbabilityFor B(n,p) , that is to say, for any

Binomial:

This is on the formula sheet!

Examples:Find the probability that exactly 3

children have type O blood. B(5,0.25)

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:

Above is the binomial from the blood type problem.

Find the expected value and standard deviation.

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 VariablesGiven B(n,p):

Remember – these formulas ONLY work for binomial distributions!

Both of these are on the formula sheet as well.

Examples Continued:Together, let’s do numbers Page 403: 69-

72

Using the first one as an example B(20, 0.85) find:

Homework (same as before)Pg. 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.

(independence)

Example:Suppose 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.

Check the conditions for using a Normal approximation in this setting.

Example Continued:Use a Normal distribution to

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

Find the meanFind the standard deviationUse the Normal curve

Homework #3 (again)Pg. 403 (73-75, 77, 79, 80, 82, 84-87,

89-92, 94-105)