# Mixed Binomial, Geometric and Random Variable Practice

AP Statistics

## Mixed Binomial, Geometric and Random Variable Practice

1.

Consider a small ferry that can accommodate cars and buses. The toll for cars is \$3 and the toll for buses is \$10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip.

Suppose the X and Y are independent and have the following probability distributions:

X 0 1 2 3 4 5

P X ) .

05 .

10 .

25 .

30 .

20 .

10

Y 0 1 2

P Y 50 .

30 .

20 a.

Compute the mean, variance and standard deviation of the number of cars on the ferry. 2.8,

1.66, 1.29

b.

Compute the mean and variance of the number of buses on the ferry. .7, .61

c.

Compute the mean and variance of total number of vehicles on the ferry. 3.5, 2.27

d.

Compute the mean and variance of the total amount of money collected on the ferry. \$15.40,

75.94

e.

Which type of vehicle is bringing in more money? How much more should we expect from that vehicle per trip? What is the variance of this amount? Cards, \$1.40, 75.94

f.

If 10 ferry rides were randomly selected, what’s the probability that exactly 3 out of 10 ferries will have exactly 2 cars? .2503

g.

If ferry rides are randomly selected, what’s the probability that it will take 4 selections to find the first ferry with two busses. .1024

2.

According to a recent Census Bureau report, 12.7% of Americans live below the poverty level.

Suppose you plan to sample at random 100 Americans and count the number of people who live below the poverty level. a.

What is the probability that you count exactly 10 in poverty? .0928

b.

What is the probability that you count 10 or less in poverty? .2614

c.

What is the probability that you start taking the random sample and you find the first person in poverty on the 8 th

person selected? .0491

3.

If you roll a pair of dice, what are the chances of getting doubles for the first time on the 5 th

roll?

.0804

4.

Dr. Fidgit developed a test to measure boredom tolerance. He administered it to a group of 20,000 adults between the ages of 25 and 35. The possible scores were 0, 1, 2, 3, 4, 5, and 6, with 6 indicating the highest tolerance to boredom. The following is the probability distribution:

X 0 1 2 3 4 5 6

P X .

.

.

.

.

.

a) Find P X

) .1 b) Find P X

) .38 c) Find P ( 1

X

4 ) .7

5.

A potential buyer will sample videotapes from a large lot of new videotapes. If she finds at least one defective one, she’ll reject the entire lot. If ten percent of the lot is defective, what is the probability that she’ll find a defective tape by the 4 th videotape? .3439

6.

“One in 10 high school graduates in the state of Florida sends an application to the University of

Florida,” says the direct of admissions. If Florida has 120,000 high school graduates next year, how many applicants would they expect? 12,000 students

7.

Michael Jordan, Allen Iverson, and Vince Carter will have their picture put in the new boxes of

Wheaties. a.

Create a distribution of the chances of landing a MJ poster up to the 5 th

box.

.3333 .2222 .1482 .0987 .0658

b.

How will the distribution appear different if the Allen Iverson poster is discontinued?

1 st

value is larger (.5), but following values will drop much quicker c.

How will the distribution appear different if Kobe Bryant, Shaquille O’Neal and Tim Duncan are added? 1 st

value is smaller (.25), but following values will drop slower

8.

A certain golfer makes her putts 60% of the time. a.

If she putts 10 times, what is the probability that she will make half or less? .3669

b.

What is the probability that the 3 rd

putt was the first one she made? .096

c.

If she putts 8 times, what is the probability that she will make 5 or more putts? .5941

d.

What is the probability that the 4 th

putt was the first one she misses? .0864

9.

A game is played with a spinner on a circle, like the minute hand on a clock. The circle is marked evenly from 1 to 100. The player spins the spinner and the resulting number is the number of seconds the player is given to solve a randomly selected mathematics problem. Suppose there are 30 students playing in the class. a.

What is the probability that 10 of the students received over a minute to solve the problem?

.1152

b.

What is the probability that exactly half of the students received 30 or less seconds?

.0106

c.

What is the probability that the fourth student was the first to receive a minute or less?

.0384

d.

What is the probability that at least half of the students received 45 seconds or more?

.9699

e.

What is the probability that a minute or more was received by the 10 th

student?

.9940

10.

Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose that a computer manufacturer receives computer boards in lots of five. Two boards are selected from each lot for inspection. Each possible outcome is a pair of numbers. a.

List the ten possible outcomes. b.

Suppose boards 1 and 2 are the only defective boards in a lot of five. Define X as the number of defective boards observed among those inspected. Find the probability distribution.