Search topics

Search

Conditional probability from exact representations

Start a set of 8
Step 5 of 5
Total equally likely outcomes
\(\displaystyle n(S)=30\)
Event A
\(\displaystyle n(A)=12\)
Event B
\(\displaystyle n(B)=15\)
Overlap
\(\displaystyle n(A\cap B)=6\)

Which calculation uses the correct condition?

Year 12, VCE Units 3 & 4, Chapter 11, 4 difficulty levels, Calculator allowedBuild a worksheet

Conditional probability

A conditional probability restricts attention to outcomes where the conditioning event is known. The event after the vertical bar supplies the denominator, and the intersection supplies the numerator

\[\Pr(A\mid B)=\frac{\Pr(A\cap B)}{\Pr(B)}\]
  1. Read after the bar In Pr(A|B), event B is the known condition and the reduced sample space \(\displaystyle \Pr(A\mid B)\text{ uses }B\)
  2. Use the intersection The numerator contains outcomes that satisfy both the required event and the condition \(\displaystyle \Pr(A\cap B)\)
  3. Use the condition denominator Divide by the probability or count of the conditioning event, not the whole space \(\displaystyle \Pr(A\mid B)=\frac{\Pr(A\cap B)}{\Pr(B)}\)
  4. Keep directions separate Reversing the events usually changes the reduced sample space and the answer \(\displaystyle \Pr(A\mid B)\ne\Pr(B\mid A)\text{ in general}\)

Worked example

A finite experiment has 30 equally likely outcomes. Event A contains 12 outcomes, event B contains 15 outcomes, and 6 outcomes are in both events. Find the probability of A given B

\[\Pr(A\mid B)=\frac{n(A\cap B)}{n(B)}\]
  1. \(\displaystyle \Pr(A\mid B)\text{ has condition }B\)The event after the bar is the reduced sample space
  2. \(\displaystyle n(A\cap B)=6,\quad n(B)=15\)Use the intersection for the numerator and the known event B for the denominator
  3. \(\displaystyle \Pr(A\mid B)=\frac{6}{15}=\frac{2}{5}=0.4\)Divide the intersection count by the count in the conditioning event and reduce

Common mistakes

Reversing the condition
✕\(\displaystyle \Pr(A\mid B)=\frac{6}{12}=\frac{1}{2}\)✓\(\displaystyle \Pr(A\mid B)=\frac{6}{15}=\frac{2}{5}\)
The event after the bar is B, so the denominator must count B rather than A.
Using the whole space as denominator
✕\(\displaystyle \Pr(A\mid B)=\frac{6}{30}=\frac{1}{5}\)✓\(\displaystyle \Pr(A\mid B)=\frac{6}{15}=\frac{2}{5}\)
A conditional probability uses the known event as the reduced sample space, not all outcomes
Using a marginal numerator
✕\(\displaystyle \Pr(A\mid B)=\frac{12}{15}=\frac{4}{5}\)✓\(\displaystyle \Pr(A\mid B)=\frac{6}{15}=\frac{2}{5}\)
The numerator must contain outcomes in both A and B, not every outcome in A.

Difficulty levels

introReduce an equiprobable listed space to the outcomes in the conditioning event
standardCalculate from a completed count or probability table
challengeCalculate both directions or recover an intersection from a stated relation
examChoose the correct formula orientation and calculate an exact conditional probability

Questions about this topic

What does the vertical bar mean in conditional probability?+

It means that B is known. The calculation is the probability of A and B divided by the probability of B.

Why is the numerator an intersection?+

The required outcome must satisfy A and the known condition B at the same time, so it belongs to the intersection of A and B

Why are Pr(A|B) and Pr(B|A) usually different?+

They use different reduced sample spaces. The first divides by the size of B and the second divides by the size of A.

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 11: Binomial distribution

In this chapter

Bernoulli trials and the Bernoulli distribution11A Bernoulli trials and the Bernoulli distribution: Finite events, probability assignments and complementsSolve a quadratic exactly with the quadratic formula11A Bernoulli trials and the Bernoulli distribution: Round to decimal places and significant figures11B Binomial distribution and parameters: Binomial conditions, the distribution table and graph shapeMean, variance, standard deviation and parameter recovery of a binomial variableExact probability of repeated independent successesEvaluate factorial and combination expressionsSolve a linear equation11B Binomial distribution and parameters: Round to decimal places and significant figuresCumulative, at-least-one and conditional probabilities of a binomial variableSmallest trial count for an at-least-one threshold11C Binomial distribution events: Finite events, probability assignments and complementsConditional probability from exact representationsSolve and interpret a linear inequation11D Binomial shape and specified order: Binomial conditions, the distribution table and graph shapeProbabilities of sequences of independent eventsDescribe the shape of a distribution