CBSE Board Examination 2023 · Section A · 1 mark

Q12: $P(B/A)$ from $P(A/B) = 0.3$ ⁠, $P(A) = 0.4$ ⁠, $P(B) = 0.8$

Board 2023Ch 13 · Probability

The question

If $P\left(\frac{A}{B}\right) = 0.3$ ⁠, $P(A) = 0.4$ and $P(B) = 0.8$ ⁠, then $P\left(\frac{B}{A}\right)$ is equal to :

(a) $0.6$

(b) $0.3$

(d) $0.4$

Exam answer

By the multiplication rule,

\begin{aligned} P(A \cap B) &= P\left(\frac{A}{B}\right) P(B) \\ &= 0.3 \times 0.8 = 0.24 \end{aligned}

Why?

Conditional probability is defined by

P\left(\frac{A}{B}\right) = \frac{P(A \cap B)}{P(B)}

Multiplying both sides by $P(B)$ gives the line above. We need $P(A \cap B)$ because it links the two conditional probabilities.

The product: $3 \times 8 = 24$ ⁠, and there are two decimal places in total (one in 0.3, one in 0.8), so $0.3 \times 0.8 = 0.24$ ⁠.

\begin{aligned} P\left(\frac{B}{A}\right) &= \frac{P(A \cap B)}{P(A)} \\ &= \frac{0.24}{0.4} = 0.6 \end{aligned}

Why?

Now the condition is $A$ ⁠, so divide by $P(A) = 0.4$ ⁠, not by $P(B)$ ⁠.

\frac{0.24}{0.4} = \frac{2.4}{4} = 0.6

Watch out: $P\left(\frac{B}{A}\right)$ and $P\left(\frac{A}{B}\right)$ are different in general. Here one is 0.6 and the other is 0.3.

Answer: (a) $0.6$

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, $P(A/B) = 0.3$ and $P(B) = 0.8$ give $P(A \cap B) = 0.24$ ⁠. In $P(B/A)$ the event after the bar is $A$ ⁠, so divide by $P(A) = 0.4$ ⁠, not by $P(B)$ ⁠.

Quick lesson · Worked NCERT solutions · Interactive lesson

Found an error?

Tell us if something on this page is wrong or unclear. Send feedback on this question →

Probability in past papers: question 10 of 38 · see them all

Q12: P(B/A)P(B/A)P(B/A) from P(A/B)=0.3P(A/B) = 0.3P(A/B)=0.3⁠, P(A)=0.4P(A) = 0.4P(A)=0.4⁠, P(B)=0.8P(B) = 0.8P(B)=0.8

Q12: $P(B/A)$ from $P(A/B) = 0.3$ ⁠, $P(A) = 0.4$ ⁠, $P(B) = 0.8$