Special Discrete Distributions Chapter 6 Special Discrete Distributions

Binomial Distribution B(n,p) Each trial results in one of two mutually exclusive outcomes. (success/failure) There are a fixed number of trials Outcomes of different trials are independent The probability that a trial results in success is the same for all trials The binomial random variable x is defined as the number of successes out of the fixed number

Are these binomial distributions? Toss a coin 10 times and count the number of heads Yes Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant Two parents with genes for O and A blood types and count the number of children with blood type O No, no fixed number

Toss a 3 coins and count the number of heads Find the discrete probability distribution X 0 1 2 3 P(x) .125 .375 .375 .125 Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

Binomial Formula: Where:

Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

The number of inaccurate gauges in a group of four is a binomial random variable. If the probability of a defect is 0.1, what is the probability that only 1 is defective? More than 1 is defective?

Calculator Binomialpdf(n,p,x) – this calculates the probability of a single binomial P(x = k) Binomialcdf(n,p,x) – this calculates the cumulative probabilities from P(0) to P(k) OR P(X < k)

A genetic trait of one family manifests itself in 25% of the offspring A genetic trait of one family manifests itself in 25% of the offspring. If eight offspring are randomly selected, find the probability that the trait will appear in exactly three of them. At least 5?

P(x < 6) = binomcdf(10,.3,6) = .9894 In a certain county, 30% of the voters are Republicans. If ten voters are selected at random, find the probability that no more than six of them will be Republicans. P(x < 6) = binomcdf(10,.3,6) = .9894 What is the probability that at least 7 are not Republicans? P(x > 7) = 1 - binomcdf(10,.7,6) = .6496

Binomial formulas for mean and standard deviation

In a certain county, 30% of the voters are Republicans In a certain county, 30% of the voters are Republicans. How many Republicans would you expect in ten randomly selected voters? What is the standard deviation for this distribution? expect

Geometric Distributions: There are two mutually exclusive outcomes Each trial is independent of the others The probability of success remains constant for each trial. The random variable x is the number of trials UNTIL the FIRST success occurs. So what are the possible values of X How far will this go? To infinity X 1 2 3 4 . . .

Differences between binomial & geometric distributions The difference between binomial and geometric properties is that there is NOT a fixed number of trials in geometric distributions!

Other differences: Binomial random variables start with 0 while geometric random variables start with 1 Binomial distributions are finite, while geometric distributions are infinite

Not on formula sheet – they will be given on quiz or test Geometric Formulas: Not on formula sheet – they will be given on quiz or test

What are the values for these random variables? Count the number of boys in a family of four children. What are the values for these random variables? Binomial: X 0 1 2 3 4 Count children until first son is born Geometric: X 1 2 3 4 . . .

No “n” because there is no fixed number! Calculator geometpdf(p,x) – finds the geometric probability for P(X = k) Geometcdf(p,x) – finds the cumulative probability for P( X < k) P(X > k) = 1- geometcdf(p,x-1) No “n” because there is no fixed number!

What is the probability that the first son is the fourth child born? What is the probability that the first son is born is at most four children?

A real estate agent shows a house to prospective buyers A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person is 35%. What is the probability that the agent will sell the house to the third person she shows it to? How many prospective buyers does she expect to show the house to before someone buys the house?