## 1 Introduction to Probability and Counting

### 1.1 Heuristic Probabilities

Idea: Assign value between 0 and 1 to event; magnitude gives likelihood event will occur
• near 0: unlikely
• near 1: likely
• near 1/2: may or may not occur (either equally likely)
How to assign probabilities
• Personal approach: guess!
• requires experience; often used when have no previous data
• ex: estimating the probability a totally new aircraft design will crash on its first flight

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• Relative frequency approach:
• conduct experiment many times; then

• P  =  m/n,   where
n = total number of times experiment is conducted
m = number of times in which desired phenomenon occurs
• requires ability to repeat
• ex: weather forecast

• If it rains on 15 out of 50 days with identical meteorological conditions, then the probability of precipitation for a day with those conditions is
P  = 15/50  =  .30  =  30%.

• Classical Approach:
• - compute total number of possible outcomes, n(S)

• - compute number of outcomes with desired result A, n(A)
- then probability P = n(A)/n(S)
• valid only if outcomes are equally likely!
• ex: roll 1 die; what's probability get an even number?

• n(S) = number of possible values = 6
n(A) = number of values with desired property (even) = 3
probability P = n(A)/n(S) = 3/6 = 1/2.
Venn diagrams can help!