medium difficulty8 min

Conditional Probability: Probability After Given Information

For two events A and B with P(B) > 0, the conditional probability of A given B is P(A|B) = P(A∩B)/P(B). It measures the chance of A after restricting the sample space to B.

In one line: After the bar means already known; the denominator is the probability of what is after the bar.

Practice This Concept

Understand it

The event written after the vertical bar is the given event. In P(A|B), B has already occurred, so only outcomes in B remain relevant. The numerator P(A∩B) counts outcomes where both A and B occur, and the denominator P(B) rescales the probability within the given event.

Example

If a card is drawn from a deck, let A be the event that the card is a king and B be the event that the card is a face card. Since there are 12 face cards and 4 kings, P(A|B) = P(A∩B)/P(B) = (4/52)/(12/52) = 1/3.

Formula and rule

Key formula: P(A|B) = P(A∩B)/P(B), valid only when P(B) > 0.

Similarly, P(B|A) = P(A∩B)/P(A), valid only when P(A) > 0.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Given P(A) = 0.6, P(B) = 0.5, and P(A∪B) = 0.8, find P(A|B).

1First use P(A∪B) = P(A)+P(B)-P(A∩B).
2So 0.8 = 0.6+0.5-P(A∩B), giving P(A∩B) = 0.3.
3Therefore P(A|B) = 0.3/0.5 = 0.6.

Method you can reuse

1Identify the required conditional probability.
2Mark the given event as the denominator.
3Find P(A∩B) from data, counting, or the union formula.
4Substitute into P(A|B) = P(A∩B)/P(B).
5Check that the denominator is not zero.

Visual tip

A Venn diagram can help show A∩B inside B, but the formula does not require a diagram. If drawn, shade the overlap A∩B and treat B as the reduced sample space.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If P(A∩B) = 0.18 and P(B) = 0.45, find P(A|B).

Think first, then show answer

P(A|B) = P(A∩B)/P(B) = 0.18/0.45 = 0.4.

Common wrong answer

Writing P(A|B) = P(B)/P(A∩B) or placing P(A) in the denominator just because A is mentioned first.

Don't mix this up

Students often interchange P(A|B) and P(B|A). These are usually different because the given event changes the sample space.

Exam tip

Before substituting, underline the phrase after 'given that'. That event must go in the denominator of the conditional probability formula.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

It appears as direct substitution, Venn-diagram based probability, card or dice situations, and word problems using phrases like 'given that', 'known that', or 'among those'.

1
Enough for 1 mark
For two events A and B with P(B) > 0, the conditional probability of A given B is P(A|B) = P(A∩B)/P(B). It measures the chance of A after restricting the sample space to B.
2
Add for 2 marks
Example: If a card is drawn from a deck, let A be the event that the card is a king and B be the event that the card is a face card. Since there are 12 face cards and 4 kings, P(A|B) = P(A∩B)/P(B) = (4/52)/(12/52) = 1/3.
3
Add for 3 marks
The event written after the vertical bar is the given event.

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