Multiplication Theorem of Probability: Calculating Probabilities of Independent Events

Learn about the multiplication theorem of probability for calculating the probability of two independent events both occurring. This guide explains the theorem, provides examples, and demonstrates its application in solving probability problems involving independent events.

Multiplication Theorem of Probability

Statement of the Theorem

The multiplication theorem states: If A and B are two independent events (meaning the occurrence of one event doesn't affect the probability of the other), then the probability that both A and B will occur is the product of their individual probabilities.

P(A ∩ B) = P(A) * P(B)

Proof of the Multiplication Theorem

Let's say:

  • There are n₁ total equally likely ways for event A to occur, with p of those being successful outcomes.
  • There are n₂ total equally likely ways for event B to occur, with q of those being successful outcomes.

The number of ways for both A and B to succeed is p * q (since the events are independent). The total number of possible outcomes is n₁ * n₂. Therefore, the probability of both A and B succeeding is (p * q) / (n₁ * n₂). Since P(A) = p/n₁ and P(B) = q/n₂, we have P(A ∩ B) = P(A) * P(B).

Extending to Multiple Events

The multiplication theorem extends to more than two independent events. For three independent events A, B, and C:

P(A ∩ B ∩ C) = P(A) * P(B) * P(C)

And, in general, for n independent events:

P(A₁ ∩ A₂ ∩ ... ∩ An) = P(A₁) * P(A₂) * ... * P(An)

Example: Drawing Balls from a Bag

A bag contains 5 green and 7 red balls. Two balls are drawn at random (without replacement—this means the events are dependent, not independent!). What's the probability of drawing one green ball and one red ball?

Solution:

(The solution, showing the calculation of P(A) - probability of drawing a green ball, P(B) - probability of drawing a red ball, and P(A∩B) - probability of drawing one green and one red ball, should be included here. Note that this example incorrectly uses the multiplication theorem for independent events, even though the events are dependent. The correct solution would require considering conditional probability.)

Conclusion

The multiplication theorem provides a straightforward way to calculate the probability of multiple independent events occurring. It's crucial to remember that this theorem applies only when the events are independent; otherwise, we need to use conditional probabilities.