Unit 4-Lecture 1：Intro to Discrete Probability

1 The Four Step Method to Solve Problems like “What is the probability that… ?”

Step 1: Find the Sample Space:

• Every possible combination of these randomly-determined quantities is called an outcome.
• The set of all possible outcomes is called the sample space for the experiment.
• A tree diagram is a graphical tool that can help us work through the four step approach when the number of outcomes is not too large or the problem is nicely structured.
• The leaves of the tree represent outcomes of the experiment, and the set of all leaves represents the sample space.

Step 2: Define Events of Interest
A set of outcomes is called an event.
Step 3: Determine Outcome Probabilities

• Step 3a: Assign Edge Probabilities
• Step 3b: Compute Outcome Probabilities
  Specifying the probability of each outcome amounts to defining a function that maps each outcome to a probability.

Step 4: Compute Event Probabilities

2 The Strange Die

There are arbitrarily large sets of dice which will beat each other in any desired pattern according to how many times the dice are rolled.

3 The Birthday Principle

If there are d days in a year and 2d−−√ people in a room, then the probability that two share a birthday is about 1−1/e=0.632

4 Set Theory and Probability

• A countable sample space S is a nonempty countable set.
• An element ω∈S is called an outcome.
• A subset of S is called an event.
• A probability function on a sample space S is a total function.
  Pr S→R such that:
  
  • Pr[ω]≥0for allω∈S and
  • ∑ω∈SPr[ω]=1

Probability Rules from Set Theory:

Sum Rule:
Pr[⋃n∈NEn]=∑n∈NPr[En]
Union Rule:
Pr[E1∪E2...∪En∪...]≤Pr[E1]+...+Pr[En]+...

Uniform Probability Spaces:
A finite probability space, S, is said to be uniform if P[ω] is the same for every outcome ω∈S.

---

Reference

[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.