1. Introduction to Discrete Probability
ICS 6D
Sandy Irani

2. A Few Applications of Discrete Probability in Computer Science
Analyzing the behavior of algorithms that make random choices
Running time, performance
Testing computer systems
Generating input/demand to test a system
Modeling discrete structures
Understanding the structure of the internet or social networks

3. Basic Definitions
Experiment: repeatable process that results in one out of a possible set of outcomes.
Sample space: the set of all possible outcomes of an experiment
Examples:
5-card hand from a standard deck
Roll of a blue and a red die
Flip a coin 3 times

4. Events An event is a subset of the sample space Examples:
Experiment: 5-card hand from a standard deck
Event: The hand has a pair of 8’s
Experiment: Roll of a blue and a red die
Event: The two dice have the same number
Experiment: Flip a coin 3 times
Event: there are more heads than tails.

5. Probabilities
Probability over the outcomes of an experiment with sample space S
p: S → ℝ such that
For every x ∈ S, 0 ≤ p(x) ≤ 1
x ∈ S 𝑝(𝑥) = 1
Example: Roll of a single die.
S = {1, 2, 3, 4, 5, 6}
p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = 2 7

6. Probability of an Event
Prob(E) = 𝑥 ∈𝐸 𝑝(𝑥)
Example: Roll of a single die.
S = {1, 2, 3, 4, 5, 6}
p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = 2 7
Event: outcome is even

7. Uniform Distribution Experiment with sample space S
The uniform distribution over S is
For every x ∈ S, 𝑝 𝑥 = 1 |𝑆|
If the distribution is uniform then:
p(E) =

8. Probabilities Under the Uniform Distribution
Roll a red and a blue die
(Fair dice means that the distribution over the outcomes is uniform)
What is |S|?
What is the probability of each event:
B: the sum of the dice is at most 3

9. Probabilities Under the Uniform Distribution
Roll a fair red and blue die
What is the probability of each event:
C: The value on the red die is one more than the value on the blue die

10. Probabilities Under the Uniform Distribution
Roll a fair red and blue die
What is the probability of each event:
D: The value on the red die is less than the value on the blue die

11. Probabilities Under the Uniform Distribution
Flip a fair coin 3 times
(Fair coin means that the distribution over the outcomes is uniform)
What is |S|?
What is the probability of each event:
A: the first and last flip come up heads

12. Probabilities Under the Uniform Distribution
Flip a fair coin 3 times
What is the probability of each event:
E: At least one flip comes up heads.

13. Probabilities Under the Uniform Distribution
Deal a 5-card hand from a perfectly shuffled deck.
(“Perfectly shuffled deck” means that the distribution over the outcomes is uniform)
What is |S|?

14. Probabilities Under the Uniform Distribution
Deal a 5-card hand from a perfectly shuffled deck.
What is the probability of each event:
F: the hand is a full house

15. Probabilities Under the Uniform Distribution
Deal a 5-card hand from a perfectly shuffled deck.
What is the probability of each event:
T: the hand has a 2-of-a-kind

16. A network has 30 servers. A copy of a file is stored on 3 of the servers. A random subset of 5 of the servers go down. What is the probability that all three copies of the file are unavailable?

17. A group of 90 kids take a test. There are three versions of the test
A group of 90 kids take a test. There are three versions of the test. The kids are grouped randomly into three groups of 30 and each group is assigned to a version of the test. What is the probability that two particular students are given the same version of the test?

18. A 7-bit string is selected at random
A 7-bit string is selected at random. What is the probability that the string is a palindrome?

19. A string over the alphabet {a, b, c, d} is selected at random
A string over the alphabet {a, b, c, d} is selected at random. What is the probability that the string has two consecutive characters that are the same somewhere in the string?

20. 8 different kids are assigned 8 different jobs
8 different kids are assigned 8 different jobs. The jobs are assigned at random so that each kid gets exactly one job. One of the kids (Barnaby) has a favorite job. What is the probability that Barnaby is assigned his favorite job?