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Last Updated: Mar 27, 2024

Multiplication Theorem

Author Akash Nagpal
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Multiplication Theorem derives a relationship between two events. Considering two events, let’s say A and B are associated within a Sample space S. Therefore AB denotes the events in which both events have simultaneously occurred. This theorem is also known as the Multiplication Rule of Probability. So, this is the core idea behind the probability multiplication theorem. Let's get to know it a little better.


Considering A and B as two independent events, the probability of both occurring simultaneously would be equal to the product of their individual probabilities.

  • For independent events,
    P(A∩B) = P(A) × P(B)
  • For dependent events, 

P(A ∩ B) = P(A) × P(B|A)

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Let us assume that, 

Event A can happen in n1 ways out of which p are successful events 

Event B can happen in n2 ways out of which q are successful events 

Therefore, the number of successful cases = combining the successful events of p and q

Thus, Number of successful cases = p × q

Total cases = n1 × n2 


P(A and B) = P(A ∩ B) = p × q ⁄ n1 × n2 = p×n1 ⁄ q×n2 

So we are having , P(A) = p ⁄ n1   and P(B) = q ⁄ n2

Hence , P(A ∩ B) = P(A) × P(B)


  1. A box contains 8 blue and 7 yellow balls. Find the probability that one is blue and the other is yellow when two balls are drawn.
    Probability of getting a blue ball = P(A) = 18 .
    Probability of getting a yellow ball = P(B) = 17
    By applying the multiplication rule for independent events, 
    we get,
    P(AB) = P(A)  P(B) = 18  17 = 156.
  2. Without replacing the first card from the deck, two cards are chosen. Calculate the chances of getting a king and then a queen.
    We have total events = 52
    Since the first draw is not replaced, the events are dependent in this case.
    P(K) = probability of selecting a king = 4/52.
    P(Q) = probability of selecting a queen = 4/51
    Since the first card drawn has not been replaced ; therefore the total events would become 51.
    Hence, P(A ∩ B) = P(A) × P(B|A) 

= 4/52  4/51 = 1/166.
Hence, the probability of selecting a king and a queen from the deck would be 1/166.


  1. What is the multiplication theorem for probability?
    According to the theorem, "the probability of simultaneous occurrence of two self-determining events is provided by the product of their individual probabilities."
  2. What is the difference between probability and conditional probability?
    Probability is concerned with the possibility of a specific occurrence occurring. Conditional probability considers the likelihood of two events occurring close to one another.

Key Takeaways

In this article, we have extensively discussed the Multiplication Theorem Of Probability with the help of examples. Check out the Addition Theorem Of Probability for the following topics.

Recommended Readings:

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