6 October 20034.1 Randomness

What does “random” mean?

4.2 Probability Models

4.5 General Probability RulesDefining random processes mathematically

Combining probabilities:

The Addition Rule

The Multiplication Rule

Conditional probabilities

Decision analysis

RANDOMNESS Random is not the same as haphazard or

helter-skelter or higgledy-piggledy.

Random events are unpredictable in the short-term, but lawful and well behaved in the long-run.

For example, if I toss one coin, I do not know whether it will land heads or tails. But if I toss a million coins, I can be reasonably certain that about half of them will be heads and the other half tails.

PROBABILITY Probabilities are numbers which describe the

outcomes of random events.

The probability of an event is the long-run relative frequency of that event.

P(A) means “the probability of event A.”

If A is certain, then P(A) = one

If A is impossible, then P(A) = zero

Sample Space A “sample space” is a list of all possible

outcomes of a random process. When I roll a die, the sample space is {1, 2, 3, 4, 5, 6}.

When I toss a coin, the sample space is {head, tail}.

An “event” is one or more members of the sample space.

For example, “head” is a possible event when I toss a coin. Or “number less than four” is a possible event when I roll a die.

Probability Rules All probabilities are between zero and one:

0 < P(A) < 1

Something has to happen:P(Sample space) = 1

The probability that something happens is one minus the probability that it doesn’t:

P(A) = 1 - P(not A)

Examples The probability that I wear a green shirt tomorrow is

some number between zero and one.

0 < P(green shirt) < 1 The probability that I wear a shirt of some color

tomorrow is equal to one.

P(shirt) = 1 The probability that I wear a green shirt tomorrow is

one minus the probability that I don’t wear one.

P(green shirt) = 1 - P(non-green shirt)

CHANCES and ODDS Chances are probabilities expressed as

percents. Chances range from 0% to 100%.For example, a probability of .75 is the same as a 75% chance.

The odds for an event is the probability that the event happens, divided by the probability that the event doesn’t happen. Odds can be any positive number.For example, a probability of .75 is the same as 3-to-1 odds.

Conditional Probability The conditional probability of B, given A, is

written as P(B|A). It is the probability of event B, given that A occurs.

For example, P(blue pants | green shirt) is the probability that I will put on a pair of blue pants, given that I have already picked out a green shirt.

Note that P(B|A) is not the same as P(A|B).

Independence Events A and B are independent if the probabiity of

event B is not affected by A’s occurring or not occurring:

If and only if A and B are independent,

P(B | A) = P(B | not A) = P(B)

For example, if I am tossing two coins, the probability that the second coin lands heads is always .50, whether or not the first coin lands heads.

P(H2 | H1) = P(H2|T1) = P(H2)

Non-independence Events A and B are not independent if P(B) is

different, depending on whether A occurs: If P(B | A) ≠ P(B | not A), then A and B are not

independent.

Suppose I don’t like to wear blue pants with a green shirt:

P(blue pants|green shirt) < P(blue pants|not-green shirt).

“Blue pants” and “green shirt” are not independent.

The Addition Rule If A and B cannot both occur, then

P(A or B) = P(A) + P(B)P(green shirt or blue shirt) = P(green shirt) + P(blue shirt)

The events “green shirt” and “blue shirt” are called disjoint.

If A and B could both occur, then

P(A or B) = P(A) + P(B) - P(A and B)P(green shirt or blue pants)

= P(green shirt) + P(blue pants) - P(green shirt and blue pants)

The probability that I wear green shirt or blue pants is the probability that I wear a green shirt PLUS the probability that I wear blue pants MINUS the probability that I wear a green shirt and blue pants.

The Multiplication Rule If A and B are independent, then

P(A and B) = P(A) x P(B)For example, if I choose my shirts and pants separately, then:

P(green shirt and blue pants) = P(green shirt) x P(blue pants)

If A and B are not independent, then

P(A and B) = P(A) x P(B | A)For example, if I choose pants that look good with my shirt, then: P(green shirt and blue pants)

= P(green shirt) x P(blue pants, given the green shirt)

A Numerical Example

SHIRT PANTS FREQUENCY

Green Blue 4

Green not Blue 6

not green Blue 36

not green not Blue 54

100

P(green shirt)

P(blue pants)

P(blue pants OR green shirt)

P(blue pants AND green shirt)

P(blue pants GIVEN green shirt)

P(blue pants GIVEN not-green shirt)

P(green shirt AND not-green shirt)

P(green shirt) = 10/100 = .1

P(blue pants) = 40/100 = .4

P(blue pants AND green shirt) = 4/100 = .04

P(blue pants OR green shirt) = .4+.1-.04 = .46

P(blue pants GIVEN green shirt) = 4/10 = .4

P(blue pants GIVEN not-green shirt) = 36/90 = .4

P(green shirt AND not-green shirt) = zero

A Numerical Example

Green shirt Not-green shirtBlue pants 4 36 40

Not-blue pants 6 54 6010 90 100

A Numerical Example

A Numerical Example

.40 Blue PantsGreen Shirt

.10 .60 Not-blue Pants

.90 .40 Blue PantsNot-green Shirt

.60 Not-blue Pants

A Numerical Example (in which shirts and pants are not independent)

SHIRT PANTS FREQUENCY

Green Blue 8

Green not Blue 2

not green Blue 32

not green not Blue 58

100

P(green shirt)

P(blue pants)

P(blue pants AND green shirt)

P(blue pants OR green shirt)

P(blue pants GIVEN green shirt)

P(blue pants GIVEN not-green shirt)

P(green shirt AND not-green shirt)

P(green shirt) = 10/100 = .1

P(blue pants) = 40/100 = .4

P(blue pants AND green shirt) = 8/100 = .08

P(blue pants OR green shirt) = .4 + .1 - .08 = .42

P(blue pants GIVEN green shirt) = 8/10 = .8

P(blue pants GIVEN not-green shirt) = 32/90 = .356

P(green shirt AND not-green shirt) = zero

A Numerical Example (in which shirts and pants are not independent)

.80 Blue PantsGreen Shirt

.10 .20 Not-blue Pants

.90 .356 Blue PantsNot-green Shirt

.644 Not-blue Pants

THE ADDITION RULE for more than two disjoint events

If A and B and C are mutually disjoint, then

P(A or B or C) = P(A) + P(B) + P(C)

P(green or blue or white shirt)

= P(green shirt) + P(blue shirt) + P(white shirt)

THE MULTIPLICATION RULE for more than two independent events

If A and B and C are mutually independent, then

P(A and B and C) = P(A) x P(B) x P(C)

If I pick shirts, pants, and belts independently:P(green shirt and blue pants and black belt)

= P(green shirt) x P(blue pants) x P(black belt)

Homework 5

4.1 (1 or 2 or 3), 8

4.2 11, 14, 20, 28

4.5 92, 96, 105