Introduction to Discrete Probability ICS 6 D Sandy

Introduction to Discrete Probability ICS 6 D Sandy Irani

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

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

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.

Probabilities •

Probability of an Event •

Uniform Distribution •

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

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

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

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

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.

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|?

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

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

• 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?

• 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?

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

• 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?

• 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?