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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

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.

Rule to remember

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.

Memory hook

After the bar means already known; the denominator is the probability of what is after the bar.

Examples and method

Worked example

Given P(A) = 0.6, P(B) = 0.5, and P(A∪B) = 0.8, find P(A|B). First use P(A∪B) = P(A)+P(B)-P(A∩B). So 0.8 = 0.6+0.5-P(A∩B), giving P(A∩B) = 0.3. Therefore P(A|B) = 0.3/0.5 = 0.6.

Method to apply

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

Diagram support

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.

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'.

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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-mark answer

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. 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. 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-mark answer

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. 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. Given P(A) = 0.6, P(B) = 0.5, and P(A∪B) = 0.8, find P(A|B). First use P(A∪B) = P(A)+P(B)-P(A∩B). So 0.8 = 0.6+0.5-P(A∩B), giving P(A∩B) = 0.3. Therefore P(A|B) = 0.3/0.5 = 0.6. 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'. Writing P(A|B) = P(B)/P(A∩B) or placing P(A) in the denominator just because A is mentioned first.
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