C
CraftExam
medium difficulty7 min

Multiplication Theorem of Probability

For two events A and B, P(A∩B) = P(A)P(B|A) when P(A) > 0, and also P(A∩B) = P(B)P(A|B) when P(B) > 0.

In one line: For 'and' in stages, multiply along the path.

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Understand it

The multiplication theorem rewrites the probability of both events occurring together using one event first and the other event under that condition. It is especially useful in word problems where probabilities are given in stages.

Example

A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Probability that both are red = P(first red)P(second red | first red) = (5/8)(4/7) = 5/14.

Formula and rule

Key result: P(A∩B)=P(A)P(B|A)=P(B)P(A|B).

For three events, P(A∩B∩C)=P(A)P(B|A)P(C|A∩B), with required conditioning probabilities defined.

See it worked out

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

Worked example

A box contains 4 defective and 6 good items.

  1. 1Two items are selected without replacement.
  2. 2Find the probability both are defective.
  3. 3Let A be first defective and B be second defective.
  4. 4P(A)=4/10.
  5. 5After A, 3 defective remain out of 9, so P(B|A)=3/9.
  6. 6Thus P(A∩B)=(4/10)(3/9)=12/90=2/15.

Method you can reuse

  1. 1Define events in order.
  2. 2Decide whether the trial is with or without replacement.
  3. 3Write the probability of the first event.
  4. 4Write the probability of the next event under the previous condition.
  5. 5Multiply the branch probabilities and simplify.

Visual tip

A tree diagram is useful for successive trials or staged choices. Each branch should show the conditional probability after the previous outcome.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If P(A)=0.4 and P(B|A)=0.7, find P(A∩B).

Think first, then show answer
P(A∩B)=P(A)P(B|A)=0.4×0.7=0.28.

Common wrong answer

Using (4/10)(4/10) in a without-replacement problem, ignoring that the total and favorable counts change after the first selection.

Don't mix this up

Students sometimes multiply P(A) and P(B) directly even when the second event's probability changes after the first event.

Exam tip

Look for words such as 'both', 'and', 'successively', or 'without replacement'. These often require the multiplication theorem.

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 is commonly asked in drawing balls, selecting cards, quality control, family or committee problems, and multi-stage probability situations.
  1. 1

    Enough for 1 mark

    For two events A and B, P(A∩B) = P(A)P(B|A) when P(A) > 0, and also P(A∩B) = P(B)P(A|B) when P(B) > 0.

  2. 2

    Add for 2 marks

    Example: A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Probability that both are red = P(first red)P(second red | first red) = (5/8)(4/7) = 5/14.

  3. 3

    Add for 3 marks

    The multiplication theorem rewrites the probability of both events occurring together using one event first and the other event under that condition.

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