Special Discrete Distributions
Binomial Distribution B(n, p) • Each trial results in one of two mutually exclusive outcomes. (success/failure) • There 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? 1) Toss a coin 10 times and count the number of heads Yes 2) Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant 3) 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 P(x) 0 1 2 3 . 125 . 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)
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?
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
Binomial formulas for mean and standard deviation
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?
• Using Binomial Graphs WS… • In L 1 – seq(x, x, 0, 10) • In L 2 – binompdf(10, . 1 , L 1) • Sketch histogram on WS
What happened to the shape of the distribution as the probability of success increased? As the probability of success increases, the shape changes from being skewed right to symmetrical at p =. 5 to skewed left.
• Calculate the mean and standard deviations for each of the probabilities What do you notice? As the probability of success increase, • the means increase. • the standard deviations increase to p =. 5, then decrease. Their values are also symmetrical.
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.
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
Geometric Formulas:
Count the number of boys in a family of four children. Binomial: X 0 1 2 3 4 Count the children until the first son is born Geometric: X 1 2 3 4 . . .
What is the probability that the first son is the fourth child born? What is the probability that the first son born is at most the fourth child?
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?
• Use Geometric Graphs WS…. • In L 1 – input numbers 1 -20 • In L 2 – geometpdf(. 1, L 1) • Sketch on worksheet • Find the means & standard deviations
• Geometric distributions are skewed right and become more strongly skewed right as the probability of success increases • Mean & standard deviation of the distributions decrease as the probability of success increase
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