A.P. STATISTICS (Unit 3B – Anticipating Patterns: Random Variables and Probability Distributions)

Villarreal Handout 3B – 6

Binomial and Geometric Distributions (with TI Calculators)

Means and Standard Deviations

Warm Up: Suppose that 38% of California drivers are voluntary organ donors. A newspaper surveys a random sample of 50 California drivers for an article on safe driving and automobile accident prevention.

(a) What is the probability that exactly 19 of the drivers surveyed are organ donors? (b) What is the probability that at least 19 of the drivers surveyed are organ donors?

As you can see from the warm up problem, using the binomial distribution formula is fine, when finding the probability of a single binomial value (as in question (a)), but can be tedious to use when finding the probability of a larger sequence of binomial values (as in question (b)). Fortunately, we can use our calculators to help us with the more cumbersome or tedious binomial and geometric distribution problems.

Example 1 Suppose that you are a telemarketer and historically only 15% of people called by a telemarketer will remain on the phone for more than a minute. Assume that X = the number of people (out of 5) who remain on phone for more than a minute. X is a binomial random variable with number of trials 5 and probability of success 0.15.

(a) What is the probability that 1 of the next 5 people you call will remain on the phone with you for more than one minute? Show proper responses using formula or calculator.

Using the TI-83 for Binomial Calculations. Go to DISTRIBUTION menu (DISTR = 2nd VARS). Scroll down to (0).

Dist: binompdf (𝑛, 𝜋, 𝑘) [Gives the probability that there will be k successes in n trials.] Note: “pdf” stands for probability density function

(b) Construct the probability distribution for random variable X.

Dist: binompdf (𝑛, 𝜋) [Gives the probability of each possible value of k (0, 1, …, n)]

(c) Find the probability that at most one of the callers will remain on the phone for more than a minute.

Go to DISTRIBUTION menu. Scroll down to (A).

Dist: binomcdf (𝑛, 𝜋, 𝑘) [Gives the probability that the number of successes is k] Note: “cdf” stands for cumulative-probability density function

Example 2 Suppose you roll a single die 20 times and are interested in how many times the outcome is “one.” (a) What is the probability that you get at least 6 “ones”? (b) What is the probability you get at least 2, but at 5 “ones”?

Like other variables we have studied, it is important to know the shape of the distribution of a random variable. Knowing the mean and standard deviation of the random variable gives us quick insight into its shape. (How?)

The mean and standard deviation of a binomial random variable: If X is a binomial random variable with number of trials n, and probability of success 𝜋, then

𝜇𝑋 = 𝑛 ∙ 𝜋 and 𝜎𝑋 = √𝑛𝜋(1 − 𝜋)

Example 3 Suppose that 75% of all customers at a gas station choose 87-octane gasoline. You observe the next 80 customers at the gas station. (Note: make sure to define the variable and distribution.) (a) How many customers do you expect to choose 87 octane gas? (b) What is the std. deviation of the number of customers that choose 87 octane gas? (c) What is the probability that at least 80% of customers choose 87 octane gas? (d) What is the probability that the number of customers who choose 87 octane gas is within one

std. deviation from the mean? Mean of a Geometric Distribution

If the probability of winning any prize on a lottery-scratcher is 0.2, then we would expect to win a prize every 5 lottery-scratchers that we played. Thus, we should expect that it we would have to play 5 lottery-scratchers, on average, until we find one that wins a prize. Notice that 5 is the reciprocal of 0.2

If 𝑋~𝐺(𝜋), 𝑡ℎ𝑒𝑛 𝜇𝑋 =1

𝜋

Example 4 Suppose that 30% of LHS students own their own graphing calculator. (a) How many LHS students, on average, would you need to randomly select in order to find one

with a graphing calculator? (b) What is the probability that it takes longer than average to select a student with a graphing

calculator? CW Practice Set (3B-6) (3B-6) Calculator/Notation Usage Practice WS