SAT probability of two events questions

What is a joint probability?

The probability of two events occurring together, denoted as P(A and B).

True or false: P(A or B) = P(A) + P(B)

False, because this only holds if A and B are mutually exclusive.

Fill in the blank: P(A and B) = P(A) * _____

P(B|A), the conditional probability of B given A.

Difference between independent and dependent events.

Independent events: P(A and B) = P(A) * P(B). Dependent events: P(A and B) = P(A) * P(B|A).

What does 'mutually exclusive' mean?

Two events cannot occur at the same time; P(A and B) = 0.

Scenario: A store has 30 blue shirts and 20 red shirts. P(blue)?

P(blue) = 30/(30+20) = 0.6.

True or false: P(A and B) can exceed P(A).

False, because P(A and B) ≤ P(A).

What is P(A or B)?

P(A or B) = P(A) + P(B) - P(A and B).

Scenario: You flip a coin and roll a die. What's P(heads and 3)?

P(heads) = 0.5; P(3) = 1/6; P(heads and 3) = 0.5 * 1/6 = 1/12.

What do you need for P(A|B)?

You need P(A), P(B), and P(A and B) to calculate it.

Fill in the blank: P(A|B) = P(A and B) / _____

P(B), the probability of B occurring.

Scenario: A bag has 4 red and 6 green marbles. P(red and green)?

P(red) * P(green) = (4/10) * (6/10) = 24/100 = 0.24.

True or false: If events A and B are independent, P(A|B) = P(A).

True, because the occurrence of B does not affect A.

Difference between P(A and B) and P(A or B).

P(A and B) is both events occurring; P(A or B) is at least one occurring.

Scenario: A dice roll and a coin flip. What's P(2 or heads)?

P(2) = 1/6; P(heads) = 1/2; P(2 or heads) = P(2) + P(heads) - P(2 and heads).

What is conditional probability?

The probability of event A occurring given that event B has occurred, denoted P(A|B).

Scenario: A class has 10 boys and 15 girls. P(boy and girl)?

Requires clarification; cannot happen simultaneously.

True or false: P(A|B) reduces uncertainty about A.

True, because knowing B occurs gives more information about A.

Scenario: A survey shows 70% like ice cream, 40% like chocolate. P(chocolate|ice cream)?

Calculate based on joint probabilities from survey data.

What is the formula for total probability?

P(A) = P(A and B) + P(A and not B).

Scenario: A box with 5 red and 5 blue balls. P(red and red)?

P(red first) = 5/10; P(red second | red first) = 4/9; P(red and red) = (5/10)(4/9) = 20/90.

Fill in the blank: If A and B are independent, P(A and B) = P(A) * _____

P(B), because their probabilities multiply.

Scenario: There are 3 cats and 2 dogs. P(cat or dog)?

P(cat) + P(dog) = 3/5 + 2/5 = 1.

True or false: P(A and A') = 0.

True, because A and A' cannot happen simultaneously.