What is the law of multiplication in probability?

The law of multiplication in probability states that the probability of two independent events occurring together is the product of their individual probabilities.

When two events A and B are independent, the probability of both events occurring together is given by the product of their individual probabilities. Mathematically, this can be expressed as:

P(A and B) = P(A) x P(B)

For example, consider the probability of rolling a 4 on a fair six-sided die and flipping a head on a fair coin. The probability of rolling a 4 is 1/6, and the probability of flipping a head is 1/2. Since these events are independent, the probability of rolling a 4 and flipping a head is:

P(rolling a 4 and flipping a head) = P(rolling a 4) x P(flipping a head)
= 1/6 x 1/2
= 1/12

The law of multiplication can also be extended to more than two events. For example, if events A, B, and C are independent, the probability of all three events occurring together is:

P(A and B and C) = P(A) x P(B) x P(C)

It is important to note that the law of multiplication only applies to independent events. If events are dependent, the probability of both events occurring together is more complicated and may involve conditional probabilities.