Conditional Probability

Conditional probability measures the probability of an event given that another event has already occurred. This updates our sample space based on new information.

Definition

Definition (Conditional Probability)

The probability of event

A

given event

B

(where

P (B) > 0

) is:

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

This formula essentially restricts the sample space to $B$ and checks what fraction of $B$ is also in $A$ .

Multiplication Rule

Rearranging the definition gives:

P (A \cap B) = P (A ∣ B) P (B)

This is useful for multi-stage experiments (Probability Trees).

Example (Drawing Cards)

Two cards are drawn from a deck without replacement. Probability both are Aces? Let

A_{1}

be "First is Ace",

A_{2}

be "Second is Ace".

P (A_{1}) = 4/52

P (A_{2} ∣ A_{1}) = 3/51

(only 3 aces left in 51 cards).

P (A_{1} \cap A_{2}) = \frac{4}{52} \times \frac{3}{51} = \frac{1}{221}

The "Two Children" Paradox

Problem: A family has two children. Given that at least one is a girl, what is the probability that both are girls?

Sample Space: ${BB, BG, GB, GG}$ (equally likely).
Event $B$ (At least one Girl): ${BG, GB, GG}$ .
Event $A$ (Both Girls): ${GG}$ .
$P (A ∣ B) = \frac{1}{3}$ . (Not 1/2!)

Practice Problems

Exercise (Problem 1)

A die is rolled. If the outcome is odd, what is the probability it is prime?

Exercise (Problem 2)

Use the multiplication rule to find the probability of drawing 3 Red cards in a row from a standard deck.

Exercise (Problem 3)

Urn A has 3 Red, 2 Blue balls. Urn B has 2 Red, 8 Blue balls. A fair coin chooses the urn. If a Red ball is drawn, what is the probability it came from Urn A? (Bayes' Theorem intro).