Conditional Probability and Independent Events

Conditional Probability if we have some information about the result…use it to adjust the probability probability value is called a “conditional probability” likelihood an event E occurs under the condition that some event F occurs notation: P(A | B ) "the probability of A, given B ".

Given They’re Male If an individual is selected at random, what is the probability a sedan owner is selected, given that the owner is male? P( sedan owner | male ) = _______?

Smaller Sample Space Given the owner is male reduces the total possible outcomes to 115. That's 40 out of 115.

In general... That is, For conditional probability, we define

In general... In terms of the probabilities, we define sedan mini-van truck totals male female P( sedan owner | male ) = _______?

Compute the probability sedan mini-van truck totals male female

Compare NOT conditional: Are Conditional:

Dependent Events? probability of owning a truck…...was affected by the knowledge the owner is male events "owns a truck" and "owner is male" are called dependent events.

Independent Events Two events E and F, are called independent if or simply the probability of E is unaffected by event F

Check Independence a single card is drawn from a deck... are the events "a spade is drawn" and "an ace is drawn" independent events? Check if P( spade and ace ) equals P(spade)P(ace) ? "drawing a spade doesn't affect the probability that an ace was drawn, an vice versa"

“Unaffected” These events are independent the given condition had no effect. That is, P(ace | spade ) = 1/13 = 4/52 = P(ace) And similarly, P(spade | ace ) = 1/4 = 13/52 = P(spade). equality is the result of the events being independent

Roll the Dice Using the elements of the sample space: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Compute the conditional probability: P( sum = 6 | a “4 was rolled” ) = ? are the events “sum = 6" and “a 4 was rolled" independent events?

Not Independent Does P(sum = 6 and a 4 was rolled) equal P(sum = 6)P(a 4 was rolled) ? Equivalently, P(sum = 6 | 4 is rolled ) = 2/11 = P(sum = 6) = 5/36 = These are dependent events.

“Affected” The events are NOT independent the given condition does have an effect. That is, P(sum = 6 | 4 is rolled ) = 2/11 = but P(sum = 6) = 5/36 = These are dependent events.

Probability of “A and B” Draw two cards in succession, without replacing the first card. P(drawing two spades) = ________? may be written equivalently as

Multiplication Rule P(1 st card is spade) P(2 nd is spade | 1 st is spade) (spade, spade)

Multiplicative Law for Probability For two events A and B, And when A and B are independent events, this identity simplifies to

Additive and Multiplicative Laws and if events A and B are mutually exclusive events, this simplifies to and if events A and B are independent events, this simplifies to

Additive law extended …

Multiplicative law extended …