Intersection Probability Calculator

P(A ∩ B) = P(A) × P(B|A)

Calculate the probability that both event A AND event B occur together.

Calculator Dice Coins Marbles Cards Learn

What is Intersection Probability?

The intersection of two events A and B, written as A ∩ B, represents all outcomes where BOTH events occur simultaneously. The probability P(A ∩ B) tells us how likely it is that both A and B happen together.

The Multiplication Rule

General formula (always works):

P(A ∩ B) = P(A) × P(B|A)

For independent events:

P(A ∩ B) = P(A) × P(B)

Key Concepts

Intersection (A ∩ B)

The set of outcomes where both A and B occur. Read as "A and B" or "A intersect B".

Independent Events

Events where the occurrence of one does not affect the probability of the other. Examples: coin flips, dice rolls (with replacement).

Dependent Events

Events where the outcome of one affects the probability of the other. Example: drawing cards without replacement.

Conditional Probability P(B|A)

The probability of B given that A has occurred. This accounts for how A changes the likelihood of B.

Example: Independent Events

Problem

You flip a fair coin and roll a fair die. What is P(Heads AND rolling a 6)?

Check independence

The coin flip does not affect the die roll - these are independent events

Find individual probabilities

P(Heads) = 1/2, P(roll 6) = 1/6

Apply multiplication rule

P(H ∩ 6) = P(H) × P(6) = 1/2 × 1/6 = 1/12

Answer

P(Heads AND 6) = 1/12 ≈ 8.33%

Example: Dependent Events

Problem

A bag has 4 red and 3 blue marbles. You draw 2 marbles without replacement. What is P(both are red)?

First draw

P(1st is red) = 4/7

Second draw (given first was red)

After removing one red: 3 red, 3 blue (6 total)

P(2nd red | 1st red) = 3/6 = 1/2

Apply general multiplication rule

P(both red) = P(1st red) × P(2nd red | 1st red) = 4/7 × 1/2 = 4/14 = 2/7

Answer

P(both marbles are red) = 2/7 ≈ 28.57%

Common Mistakes to Avoid

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Using P(A) × P(B) for dependent events - Only works when events are independent!

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Forgetting to update probabilities - After drawing without replacement, the total changes.

⚠️

Confusing intersection with union - Intersection is AND (both), union is OR (either).

Frequently Asked Questions

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What is Intersection Probability?

The Multiplication Rule

General formula (always works):

P(A ∩ B) = P(A) × P(B|A)

For independent events:

P(A ∩ B) = P(A) × P(B)

Key Concepts

Intersection (A ∩ B)

The set of outcomes where both A and B occur. Read as "A and B" or "A intersect B".

Independent Events

Events where the occurrence of one does not affect the probability of the other. Examples: coin flips, dice rolls (with replacement).

Dependent Events

Events where the outcome of one affects the probability of the other. Example: drawing cards without replacement.

Conditional Probability P(B|A)

The probability of B given that A has occurred. This accounts for how A changes the likelihood of B.

Example: Independent Events

Problem

You flip a fair coin and roll a fair die. What is P(Heads AND rolling a 6)?

Check independence

The coin flip does not affect the die roll - these are independent events

Find individual probabilities

P(Heads) = 1/2, P(roll 6) = 1/6

Apply multiplication rule

P(H ∩ 6) = P(H) × P(6) = 1/2 × 1/6 = 1/12

Answer

P(Heads AND 6) = 1/12 ≈ 8.33%

Example: Dependent Events

Problem

A bag has 4 red and 3 blue marbles. You draw 2 marbles without replacement. What is P(both are red)?

First draw

P(1st is red) = 4/7

Second draw (given first was red)

After removing one red: 3 red, 3 blue (6 total)

P(2nd red | 1st red) = 3/6 = 1/2

Apply general multiplication rule

P(both red) = P(1st red) × P(2nd red | 1st red) = 4/7 × 1/2 = 4/14 = 2/7

Answer

P(both marbles are red) = 2/7 ≈ 28.57%