Chapter 8 Review: The Binomial and Geometric Distributions

1. Chapter 8 Review:The Binomial and Geometric Distributions Kate Schwartz & LexyEllingwood

2. The Big Idea • This chapter focuses on two important classes of discrete random variables, each of which involves two outcomes or events of interest. Both require independent trails and the same probability of success on each trial. The binomial random variable requires a fixed number of trials; the geometric random variable has the property that the number of trials varies. Both the binomial and the geometric settings occur sufficiently often in applications that they deserve special attention.

3. Vocabulary You Need to Know • Mean • Standard Deviation • Normal Approximation • Geometric Distribution • Geometric Probability • Binomial Setting • Binomial Probability • Binomial Coefficient • Factorial n! • Probability Distribution Function • Cumulative Distribution Function

4. Key Topics Covered in this Chapter • Binomial • Identify a random variable as a binomial by verifying four conditions: two outcomes (success and failure); fixed number of trials; independent trials; and the same probability of success for each trial • Use technology or the formula to determine binomial probabilities and to construct probability distribution tables and histograms • Calculate cumulative distribution functions for binomial random variables, and construct cumulative distribution tables and histograms • Calculate means (expected values) and standard deviations of binomial random variables • Use Normal approximation to the binomial distribution to compute probabilities • Geometric • Identify a random variable as a geometric by verifying four conditions: two outcomes (success and failure); independent trials; the same probability of success for each trial; and the count of interest is the number of trials required to get the first success • Use technology or the formula to determine geometric probabilities and to construct probability distribution tables and histograms • Calculate cumulative distribution functions for geometric random variables, and construct cumulative distribution tables and histograms • Calculate means (expected values) and standard deviations of geometric random variables

5. Formulas You Should Know • Geometric Probability • Geometric Mean • Geometric Standard Deviation • Probability that is takes more than n trials to see the first success • Binomial Probability • Binomial Coefficient • Factorial n! • Binomial Mean • Binomial Standard Deviation • Normal Approximation

6. Calculator Key Strokes n = number of trials p = probability of success r = number of success • Binomial probability P(X = r) of exactly r successes in n independent trials, with probability of success p for a single trial. If r is omitted, gives a list of all probabilities from 0 to n binompdf(n,p,r) • Binomial cumulative probability P(X ≤ r) of r or fewer successes in n independent trials, with probability of success p for a single trial. If r is omitted, gives a list of all cumulative probabilities from 0 to n binomcdf(n,p,r) • Geometric probability P(X = n) that the first success occurs on the nth trial in a series of independent trials, with probability of success p for a single trial geometpdf(p,n) • Geometric cumulative probability P(X ≤ n) that the first success occurs on or before the nth trial in a series of independent trials, with probability of success p for a single trial geometcdf(p,n)

7. Example Problem: Binomial Probability • A board game has a spinner on a circle that has five equal sectors, numbered 1, 2, 3, 4, and 5, respectively. If a player has four spins, find the probability that the player spins an even number no more than two times on those four spins.

8. Example Problem: Binomial Probability • There are 2 even numbers out of the 5 numbers.No more than = at most.

9. Example Problem: Geometric Probability • In a standard deck of 52 cards, there are 12 face cards. So the probability of drawing a face card from a full deck is 12/52 = 0.231. 1. If you draw cards with replacement (that is, you replace the card in the deck before drawing the next card), what is the probability that the first face card you draw is the 10th card?

10. Example Problem: Geometric Probability • P(X= 10) == (1 – 0.231)9 (0.231)= 0.022. On the TI-83/84: geometpdf(0.231,10)= 0.0217)

11. Helpful Hints Geometric • The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes (r) is equal to 1. Binomial • To determine P(X = x) Use binompdf(n, p, x): where n is the number of observations, p is the probability of success. • To determine P(X ≤ x) Use binomcdf(n, p, x): where n is the number of observations, p is the probability of success. • To determine P(X > x) Use 1-binomcdf(n, p, x): where n is the number of observations, p is the probability of success. • To determine P(X < x) Use binomcdf(n, p, x-1): where n is the number of observations, p is the probability of success.