STAT 1100: Chapter 13: General Rules of Probability

Disjoint vs. independent

Disjoint events are not independent. This is because if you know A occurs, you know B doesn't occur if A and B are disjoint events and thus they are not independent because knowing A directly affects the probability that B occurs

P( A and B) for disjoint events is..

Equal to zero. This is because if two events are disjoint, they can never occur at the same time

Conditional notation explained

Event before the | is what we are trying to find. Event after the | is what we are given. You always divide by what you are given

P(dog) is 0.33 P(dog | not cat) is 0.38

Since knowing the outcome of event B (no cat) changed the probability of event A (dog), the two events are related and are dependent

Formula for conditional probability

The probability of A and B is occurring is divided by the probability of the event that has occurs

General multiplication rule

The probability that two events A and B happen together can be found by 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴) = 𝑃(𝐴 ∩ B)

Independent events

Two events A and B are said to be independent if knowing that one occurs does not change the probability that the other occurs

Two events A and B that both have positive probability are independent if

𝑃(𝐵|𝐴) = P(B)

General addition rule but mainly for independent events

-For disjoint events, P(A or B)= P(A) + P(B) -For any two independent events P(A or B)= P(A) + P(B)- P(A and B)

Conditional probability or dependent probabilty

-The conditional probability of B given A is the probability of event B given that we know event A occurs

Intersection

-The intersection of events A and B is the event that occurs when both A and B occur (overlap section of the venn diagram) -(A ∩ B)

Union

-The union of events A and B is the event that occurs when either A or B occurs -This is denoted as (A or B) = (A ∪ B)

P(A and B)= 0.08 P(A) * P(B)= 0.12

A and B are not independent

Tossing two dice: the result of the first toss has no effect on the result of the second toss

Example of independent events

Conditional probability derivation #1

If A and B are independent, then P(A ∩ B) = P(A) * P(B)

Multiplication rule for independent events

P(A and B),the intersection, is EQUAL to the value of P(A) * P(B)

Venn diagrams

Used to illustrate the sample space, outcomes, and events

Probability Trees

Useful device to calculate probabilities when using the conditional probability rules