Question

please explain the multiplication rule as it applies to the Rules of Probability in at least 400 words

Answer 1

Rule of Multiplication

The rule of multiplication applies to the situation when we want to know the probability of the intersection of two events; that is, we want to know the probability that two events (Event A and Event B) both occur.

Rule of Multiplication The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.

P(A ∩ B) = P(A) P(B|A)

Example
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?

Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:

• In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
• After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.

Therefore, based on the rule of multiplication:

P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15 = 0.133