1. Chapter 14 – From Randomness to Probability

2. Probability Around Us 20% chance of rain today
1 in 10 bottle caps wins a free soda (odds)
Will you hit traffic today on the way home?
Guessing on a multiple choice question
Getting a flush in a poker hand
Should you take out collision insurance?
Patterns/coincidences
Same artist comes up 2 songs in a row on mp3 player
2 people born on the same day

3. Types of probability Subjective Empirical Theoretical
No way to measure, just a guess/estimate
Empirical
Observed probability
Can use experiments or simple observations
Theoretical
Probabilities that can be calculated exactly

4. Terminology Trial: each occasion that a random phenomenon is observed
Outcome: result of trial
Event: combination of results
Sample Space: all possible outcomes

5. Example Roll 2 dice, look at the sum and see if it’s odd
Trial: each roll of 2 dice
Outcome: sum of each roll of 2 dice
Event: sum is odd
Sample Space: all possible rolls of 2 dice (36 in total)

6. Law of Large Numbers (LLN)
For independent trials, as the number of trials increases, the long-run relative frequency of repeated events gets closer and closer to a single value.
Relative frequency observed is empirical probability

7. Nonexistent Law of Averages
LLN only applies to long-term observations
Outcomes are not “due” to happen to even things out
Long-term observations happen over a very long time

8. Modeling Probability P(A) = Outcomes need to be equally likely!
P(heads)
P(roll a 6)
P(face card)
P(student at random is male)

9. Formal Probability All probabilities are between 0 and 1: 0 ≤ P(A) ≤ 1
Set of all possible outcomes has probability of P(S) = 1
Complement of A: Ac P(A) = 1 – P(Ac)

10. Addition Rule (simple version)
Assuming A and B are disjoint (mutually exclusive) events,
P(A or B) = P(A) + P(B)
P(2 or Q from a deck of cards)
P(4 or 5 on a single die)

11. Multiplication Rule (simple version)
For two independent events A and B, P(A and B) = P(A) x P(B)
P(flip 3 coins, 3 Heads)
P(draw 2 cards with replacement, 2 face cards)
P(draw 2 cards with replacement, neither face cards)
P(flip 3 coins, at least 1 Heads)