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Grade 11-12 (age 16-18)

high school Conditional Probability

Once you know that something has already happened, the same event can have a different probability. The conditional probability P(A|B) narrows the sample space to B, given that B has happened, and measures again what share of it A takes up; the multiplication rule and Bayes' theorem both grow from this idea. Bayes' theorem reasons backward from a result to the probability of its cause, which is why, for a rare disease, a positive test still leaves a surprisingly low chance of actually having it. In the first figure the sliders for the cells of A, B and A∩B divide a sample space of 10 cells and then pull B out and enlarge it; in the second, moving the prevalence slider changes the share of real patients among those who test positive.

How a Condition Changes Probability

🔍 What conditional probability is
①The probability of event A given that event B has happened is written P(A|B)
②Knowing the condition shrinks the sample space from all of S down to B
③So the denominator becomes P(B), and the numerator is the part of A inside B, P(A ∩ B)

Narrowing the Sample Space to B

4
5
2
Definition of conditional probability
P(A|B) = P(A ∩ B)P(B)
The probability of A when B has happened (P(B) > 0)
Multiplication rule
P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A)
The probability that both events happen is one event's probability times a conditional probability

Bayes' Theorem

1%
🏥 Why a positive result can still mean a low probability
①This test marks 95% of patients positive and 90% of healthy people negative
②At 1% prevalence, out of 1000 people the 10 patients give 9.5 positives and the 990 healthy people give 99 false positives
③Of the 108.5 positives only 9.5 are patients, so P(D|+) is about 8.8%
④When the prior probability (prevalence) is low, the probability after a positive result (the posterior) also comes out low
⑤The denominator P(+) is the total probability of both branches, P(+|D)P(D) + P(+|Dᶜ)P(Dᶜ)
Bayes' theorem
P(A|B) = P(B|A) × P(A)P(B)
Works backward from the result B to the probability of the cause A

Worked Examples

Example 1
If P(A) = 1/2 and P(A∩B) = 1/3, find the conditional probability P(B|A).
1
The condition is A, so P(A) goes in the denominator.
P(B|A) = P(A∩B)/P(A)
2
Substitute the given values and compute.
= (1/3)/(1/2) = 2/3
▸ 2/3
We want P(B|A), so the condition is A. Putting P(B) in the denominator gives the wrong answer.
Example 2
In a class of 30, 18 students wear glasses and 12 of them are boys. If one student who wears glasses is picked, what is the probability that it is a boy?
1
The condition is wearing glasses, so the sample space narrows to those 18 students.
P(boy | glasses) = (boys with glasses)/(glasses-wearers)
2
Find the share of the 18 taken up by the 12 boys.
= 12/18 = 2/3
▸ 2/3
The 30 students of the whole class are never used, because the condition changes the sample space to 18.

Summary

Key results
P(A|B) = P(A ∩ B)P(B), P(A|B) = P(B|A)P(A)P(B)
The definition of conditional probability and Bayes' theorem
Provincial mock exam type
For two events A and B with P(A) = 0.4 and P(B|A) = 0.5, what is P(A∩B)?
①0.1
②0.2
③0.3
④0.4
⑤0.5
▸ ② 0.2
1
This is the probability that both events happen, so use the multiplication rule.
P(A∩B) = P(A) × P(B|A)
2
Substitute the values and multiply.
= 0.4 × 0.5 = 0.2
🎯 Exam Points
①P(A|B) = P(A ∩ B) ÷ P(B); the event in the condition goes in the denominator
②The multiplication rule P(A ∩ B) = P(B) P(A|B) gives the probability that both happen
③If A and B are independent, P(A|B) = P(A)
④P(A|B) and P(B|A) are in general different
⑤In a tree diagram, multiply along a branch and add the branches that lead to the same outcome
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Meaning of Probability
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Independent Trials

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