Chapter 7 – Probability Distributions
Things You Should Know
1. Uniform Distributions
a) Situation where success occurs with the same probability (flipping a coin, rolling a dice,
cutting a card)
b) P (X ) 
1
n
c) E (X ) 
 xP (x )
where n is the number of different possibilities
d) Typical Question:
- A lottery consists of the following prizes: 1 first prize of $5 million, 2 second prizes of
$1 million and 5 third prizes of $100,000. If 20 million tickets are sold, what is the
expected value of each ticket?
E (x ) 
1
2
5
5, 000, 000  
1, 000, 000  
100, 000 
20, 000, 000
20, 000, 000
20, 000, 000
2. Binomial Distributions
a) Situation where repeated, independent trials take place, where the only possible outcomes
are success or failure.
b) P (x )  n Cx p x q n x Where p = prob. of success, q = prob. of failure q  1  p 
c) E (x )  np
d) Typical Questions
- A multiple choice test consists of 10 questions each with 4 possible answers. What is
the probability of getting 4 questions right?
P (x  4) 
4
10
6
C4  0.25   0.75 
- A factory produces light bulbs with a 1% defect rate. In a batch of 500 light bulbs,
what is the probability that at least 1 is defective? How many light bulbs should you
expect to be defective.
0
500
P (x  1)  1  P (x  0)  1  500 C0  0.01  0.99 
E (x )  500  0.01
3. Geometric Distributions
a) Situation where the probability of waiting time before success (failure happens first) is
calculated.
b) P (x )  q x p
c) E (x ) 
q
p
d) Typical Questions:
- What is the probability that the light bulb factory (above) must test 4 light bulbs
before they find a defective one?
4
P (x  4)   0.99   0.01 
- How many lights bulbs should you expect to test before you find a defective one?
E (x ) 
0.99
0.01
4. Hypergeometric Distributions
a) Situation where there is a pool to select from. The repeated trials are dependant on
eachother as the pool size decreases as you select from it.
b) P (x ) 
c) E (x ) 
a
Cx  n a Cr-x
n Cr
where a = number of successful outcomes
r = number of selections
n = total possibilities
ra
n
d) Typical Questions
- A committee of 5 is to be formed from 14 men and 12 women. What is the probability
that there will be three men on the committee? What is the expected number of men?
P (x  3) 
14
C3  12 C2
26 C5
E (x ) 
5 14 
26
- One summer, a conservation official caught and tagged 98 beavers in a river’s flood
plain. Later, 50 beavers were caught and 32 had been tagged. Estimate the size of the
beaver population.
32 
50  98 
n
Chapter 7 Practice
1. James has designed a board game that uses a spinner with 10 equal sectors number 1 to 10. If
the spinner stops on an odd number, the player moves forward double that number of
squares. However, if the spinner stops on an even number, the player must move back half
that number of squares. Determine the expected move per spin and use that number to
justify whether the game is fair or not.
2. Joe’s coffee prints prize coupons under the rims of 40% of its paper cups. If you buy 10 cups
of coffee,
a) What is the probability that you would win at least two prizes?
b) What is the expected number of prizes?
3. A dart board contains 20 equally sized sectors number 1 to 20. A dart is randomly tossed at
the board 10 times.
a) What is the probability that the dart lands in the sector labeled 20 twice?
b) What is the expected number of times the dart would land in a given sector?
4. Your favourite TV station has 10 minutes of commercials per hour. How long should you
expect to wait before you randomly select this channel without hitting a commercial?
5. A factory making printed-circuit boards has a defect rate of 5.6% on one of its production
lines. An inspector tests randomly selected circuit boards from this production line.
a) What is the probability that the first defective circuit board will be the fourth one
tested?
b) What is the probability that the first defective circuit board will be among the first
three tested?
6. Of the 15 students who solved the challenge question in a mathematics contest, 8 were
enrolled in Data Management. Five of the solutions are selected at random for a display.
a) What is the probability that at least 3 of the displays will be from students enrolled in
Data Management?
b) What is the expected number of solutions on display that were prepared by Data
Management students?
7. Suppose that one fifth of the cards in a scratch-and-win promotion gives a prize.
a) What is the probability that you will win a prize on your 3rd try?
b) What is the probability of winning within your first two tries?
c) What is the expected number of cards you would have to try before winning a prize?
8. In the spring, the Ministry of the Environment caught and tagged 500 raccoons in an area in
cottage country. The raccoons were released after being vaccinated against rabies. To
estimate the raccoon population in the area, the ministry caught 30 raccoons during the
summer. Of these, 25 had tags. Estimate the raccoon population in the area.