Calculate P(A|B) = P(A ∩ B) / P(B)

Quick Examples to Try:

where \( P(B) > 0 \)

Understanding the Components:

Multiplication Rule

Rearranging the conditional probability formula gives us the multiplication rule[web:27]:

Bayes' Theorem

Bayes' Theorem allows us to reverse conditional probabilities[web:27][web:33][web:39]:

This is incredibly useful when we know P(B|A) but need to find P(A|B)[web:36][web:40].

Independent Events

When events A and B are independent, the occurrence of B doesn't affect A[web:30]:

Example 1: Rolling Dice

Problem: When rolling a fair six-sided die, find the probability of rolling an even number given that the number rolled is greater than 4[web:23].

Solution:

Let A = rolling an even number {2, 4, 6}

Let B = rolling a number greater than 4 {5, 6}

A ∩ B = {6} (both even AND greater than 4)

P(A ∩ B) = 1/6

P(B) = 2/6 = 1/3

P(A|B) = P(A ∩ B) / P(B) = (1/6) / (1/3) = 1/2

Answer: The probability is 1/2 or 50%

Example 2: Medical Diagnosis

Problem: In a population, 5% of people have a disease. A diagnostic test shows positive results in 90% of people who have the disease. If someone tests positive, what is the probability they actually have the disease, assuming 10% of healthy people also test positive[web:25][web:29]?

Solution (Using Bayes' Theorem):

Let D = has disease, T = tests positive

P(D) = 0.05, P(T|D) = 0.90, P(T|D') = 0.10

P(T) = P(T|D)·P(D) + P(T|D')·P(D') = 0.90(0.05) + 0.10(0.95) = 0.045 + 0.095 = 0.14

P(D|T) = [P(T|D)·P(D)] / P(T) = (0.90 × 0.05) / 0.14 = 0.045 / 0.14

Answer: P(D|T) ≈ 0.321 or 32.1%

Example 3: Cards

Problem: From a standard deck of 52 cards, one card is drawn. What is the probability that it's a king given that it's a face card?

Solution:

Let A = drawing a king (4 kings in deck)

Let B = drawing a face card (12 face cards: J, Q, K in 4 suits)

A ∩ B = drawing a king that is also a face card = 4 cards

P(A ∩ B) = 4/52

P(B) = 12/52

P(A|B) = (4/52) / (12/52) = 4/12 = 1/3

Answer: The probability is 1/3 or approximately 33.3%

💡 Key Insight

Conditional probability helps us update our beliefs about events when we receive new information. This is the foundation of Bayesian reasoning[web:27][web:36].

💡 Common Mistake

Don't confuse P(A|B) with P(B|A)! These are generally different values. For example, P(cough|sick) ≠ P(sick|cough)[web:28].

💡 Notation

The vertical bar "|" reads as "given that" or "given." So P(A|B) is read as "the probability of A given B"[web:21][web:25].

💡 Restriction

Conditional probability is only defined when P(B) > 0. You cannot condition on an impossible event[web:21][web:22].

💡 Independence Test

Events A and B are independent if and only if P(A|B) = P(A), which means knowing B occurred doesn't change the probability of A[web:30].

💡 Historical Note

Thomas Bayes (1701-1761) first formulated what we now call Bayes' Theorem, which revolutionized how we think about conditional probability[web:27][web:36].

💡 Curriculum Coverage

Conditional probability appears in IB Math (SL & HL), AP Statistics, A-Level Mathematics, GCSE/IGCSE Statistics, and college probability courses.

Property 1: Range

Conditional probabilities always fall between 0 and 1: \( 0 \leq P(A|B) \leq 1 \)

Property 2: Certainty

If A is certain to occur when B occurs, then \( P(A|B) = 1 \)

Property 3: Complement

\( P(A'|B) = 1 - P(A|B) \) where A' is the complement of A

Property 4: Chain Rule

For multiple events[web:21]:

Problem 1

In a class of 100 students, 60 study mathematics and 40 study physics. If 25 students study both subjects, what is the probability that a randomly selected student studies mathematics given that they study physics?

Problem 2

A bag contains 5 red balls and 7 blue balls. Two balls are drawn without replacement. What is the probability that the second ball is red given that the first ball was red?

Problem 3

The probability that it rains is 0.3. The probability that you carry an umbrella is 0.4. The probability that it rains AND you carry an umbrella is 0.25. What is the probability you carry an umbrella given that it rains?

Adam

Co-Founder @RevisionTown

Math Expert in various curricula including IB, AP, GCSE, IGCSE, and more.