Statistics Chapter 5

How to remember whether unions or intersection have more.

"Or has more ... than And" Also ∪ is like a cup which holds more than ∩

What is the sample space (S)?

The sample space S of a chance process is the set of all possible outcomes.

Mutually Exclusive

- A and B together is impossible: P(A and B) = 0 - A or B is the sum of A and B: P(A or B) = P(A) + P(B) (Addition rule for mutually exclusive events)

Not Mutually Exclusive

- A or B is the sum of A and B minus A and B: P(A or B) = P(A) + P(B) − P(A and B) (General Addition Rule for Two Events)

What are the rules of probability models?

- The probability of any event is a number between 0 and 1. - All possible outcomes together must have probabilities whose sum is 1. - If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula: P(A) = number of outcomes corresponding to event A/total numbers of outcomes in sample space - If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

What is a probability model?

A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

What is an event?

An event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on.

How to remember the union and intersection symbols.

Hint: To keep the symbols straight, remember ∪ for union and ∩ for intersection.

Multiplication rule for independent events

If A and B are independent events, then the probability that A and B both occur is P(A ∩ B) = P(A) • P(B).

Complement rule

P(AC) = 1 - P(A)

Conditional probability formula

P(B|A) = P(A ∩ B) / P(A).

General multiplication rule

The probability that events A and B both occur can be found using the general multiplication rule P(A ∩ B) = P(A) • P(B | A) where P(B|A) is the conditional probability that event B occurs given that event A has already occurred.

What is conditional probability?

The probability that one event happens given that another event is already known to have happened is called a conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B | A).

What does it mean to say that two events are independent?

Two events A and B are independent if the occurrence of one event has no effect on the chance that the other event will happen. In other words, events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B).

When are two events mutually exclusive (disjoint)?

Two events are mutually exclusive (disjoint) if they have no outcomes in common and so can never occur together. - Turning left and turning right are Mutually Exclusive (you can't do both at the same time) - Tossing a coin: Heads and Tails are Mutually Exclusive - Cards: Kings and Aces are Mutually Exclusive