Probability

Probability

Probability Model

• A probability model is a description of some chance process that consists of two parts: a list of all possible outcomes and the probability for each outcome.

• The list of all possible outcomes is called the sample space.

• A probability model allows us to find the probability of an event.

• An event is any collection of outcomes from some chance process.

Example: Probability Model

Imagine rolling two fair, six-sided dice-one that’s red and one that’s blue. How do we develop a probability model for this chance process? Each of these 36 outcomes will be equally likely and have probability 1/36.

Finding probabilities

• Event A: Getting a sum of 5

• P(A) = 4/36 = 0.111

Multiple Choice

If you flip a coin 2 times, what's the probability of getting heads exactly one time?

• 1/4

• 2/4

• 3/4

• 4/4

Basic Probability Rules

• The probability of any event is a number between 0 and 1.

• All possible outcomes together must have probabilities that add up to 1.

• The probability that an event does not occur is 1 minus the probability that the event does occur.

Multiple Choice

Is 0.36 a valid probability for an event?

• Yes

• No

Multiple Choice

Is 1.6 a valid probability for an event?

• Yes

• No

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Mutually Exclusive

• Two events A and B are mutually exclusive if they can not happen at the same time. P(A and B) = 0

• The addition rule for mutually exclusive events A and B says that P(A or B) = P(A) + P(B)

2015 AP Statistics

• Find the probability that the chosen student scored less than a 3.

• P(scored less than 3) = P(scored 1 or 2) = P(scored 1) + P(scored 2) = 0.236 + 0.186 = 0.422

2015 AP Statistics

• Find the probability that the chosen student scored earned a passing score. (Passing = 3, 4, or 5)

• P(passing score) = 1 - P(less than 3) = 1 - 0.422 = 0.578

• Also: P(passing) = P(3 or 4 or 5) = P(3)+P(4)+P(5) = 0.252+0.191+0.135=0.578

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Two-Way Table

In a class of students, the following data table summarizes the gender of the students and whether they have an A in the class.

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Multiple Choice

What is the probability that a student selected at random is female?

• 4/13

• 9/13

• 13/27

• 4/27

Multiple Choice

What is the probability that a student selected at random is female and has an A?

• 4/13

• 9/13

• 13/27

• 4/27

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Conditional Probability

The probability that one event happens given that another event is know to have happened. The conditional probability that event B happens given that event A has happened is denoted by P(B | A).

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Independent Events

A and B are independent events if knowing whether or not one event has occurred does not change the probability of that the other event will happen. P(A | B) = P(A | B^c) = P(A)

Independent Events

• You pick an 8 of hearts from a deck of cards, put it back in, and pick another.

• The probability of picking an 8 or hearts will not change, so they are independent events.

Independent Events

• You pick an 8 of hearts from a deck of cards, then pick another.

• The probability of picking an 8 or hearts will be different, so they are dependent events.

Are "male" and

"left-handed" independent?

• P(left-hand | male) = 7/46 = 0.152

• P(left-hand | female) = 3/54 = 0.056

• Because the probabilities are not equal, the events "male" and "left-handed" are not independent.

Are "gender" and

"passing" independent?

• P(pass| male) = 110/165 = 0.667

• P(pass) = 164/246 = 0.667

• Because the probabilities are equal, "gender" and "passing" are independent.

Are "gender" and

"passing" independent?

• P(pass | female) = 54/81 = 0.667

• P(pass) = 164/246 = 0.667

• Because the probabilities are equal, "gender" and "passing" are independent.