Chapter 14 – From Randomness to Probability

Probability Around Us

20% chance of rain today1 in 10 bottle caps wins a free soda (odds)Will you hit traffic today on the way home?Guessing on a multiple choice questionGetting a flush in a poker handShould 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

Types of probability

Subjective No way to measure, just a guess/estimate

Empirical Observed probability Can use experiments or simple observations

Theoretical Probabilities that can be calculated exactly

Terminology

Trial: each occasion that a random phenomenon is observed

Outcome: result of trial

Event: combination of results

Sample Space: all possible outcomes

Example

Roll 2 dice, look at the sum and see if it’s odd

Trial: each roll of 2 diceOutcome: sum of each roll of 2 diceEvent: sum is oddSample Space: all possible rolls of 2 dice (36

in total)

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

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

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)

# of outcomes in A# of possible outcomes

Formal Probability

All probabilities are between 0 and 1:

0 ≤ P(A) ≤ 1

Set of all possible outcomes has probability of 1

P(S) = 1

Complement of A: Ac

P(A) = 1 – P(Ac)

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)

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)