CBSE Board Examination 2022 (Term 2) · Section A · 2 marks

Q4: $P(B/A)$ from $P(A) = \frac58$ ⁠, $P(B) = \frac12$ ⁠, $P(A/B) = \frac34$

Board 2022 T2Ch 13 · Probability

The question

Let A and B be two events such that $P(A) = \dfrac58$ ⁠, $P(B) = \dfrac12$ and $P(A/B) = \dfrac34$ ⁠. Find the value of $P(B/A)$ ⁠.

Exam answer

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

Why?

This is the definition of conditional probability. We know $P(A/B)$ and $P(B)$ ⁠, so it gives the one missing number, $P(A \cap B)$ ⁠. That number links the two conditional probabilities.

\Rightarrow \frac34 = \frac{P(A \cap B)}{\frac12}

\Rightarrow P(A \cap B) = \frac34 \times \frac12 = \frac38

Why?

Multiply both sides by $P(B) = \frac12$ to clear the denominator.

\begin{aligned} \frac34 \times \frac12 &= \frac{P(A \cap B)}{\frac12} \times \frac12 \\ \frac34 \times \frac12 &= P(A \cap B) \\[4pt] \frac34 \times \frac12 &= \frac{3 \times 1}{4 \times 2} = \frac38 \end{aligned}

Watch out: dividing by $\frac12$ on the right is undone by multiplying by $\frac12$ ⁠, not by 2.

\begin{aligned} P(B/A) &= \frac{P(B \cap A)}{P(A)} \\ &= \frac{3/8}{5/8} = \frac35 \end{aligned}

Why?

Now the condition is $A$ ⁠, so divide by $P(A) = \frac58$ ⁠. Also $B \cap A$ is the same event as $A \cap B$ ⁠.

Both fractions have denominator 8, so it cancels:

\begin{aligned} \frac{3/8}{5/8} &= \frac38 \times \frac85 \\ &= \frac{3 \times 8}{8 \times 5} \\ &= \frac{24}{40} \end{aligned}

Divide the top and the bottom by 8:

\frac{24 \div 8}{40 \div 8} = \frac35

$\therefore\ P(B/A) = \dfrac35$

Where this comes from

Chapter 13, Probability: Conditional probability.

Question type: Given numbers

How to spot it: click to see

Type 1. Given numbers: divide by the event after the bar

Spot it: the question gives $P(E)$ ⁠, $P(F)$ and $P(E\cap F)$ ⁠, and asks for $P(E|F)$ or $P(F|E)$ ⁠.

Here: First, $P(A/B) = \frac34$ and $P(B) = \frac12$ give $P(A \cap B) = \frac38$ ⁠. In $P(B/A)$ the event after the bar is $A$ ⁠, so divide by $P(A) = \frac58$ ⁠.

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Probability in past papers: question 13 of 38 · see them all

Q4: P(B/A)P(B/A)P(B/A) from P(A)=58P(A) = \frac58P(A)=85​⁠, P(B)=12P(B) = \frac12P(B)=21​⁠, P(A/B)=34P(A/B) = \frac34P(A/B)=43​

Q4: $P(B/A)$ from $P(A) = \frac58$ ⁠, $P(B) = \frac12$ ⁠, $P(A/B) = \frac34$