4.4 Conditional probability and Independence

4.4 Conditional probability and Independence

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.