In this short lesson we would understand how to use the Multiplication Law in Conditional Probability.
You may want to review Basics of Conditional Probability.
The Multiplication Law helps us to find conditional probability and is given by the formula:
P(A ∩ B) = P(B | A) P(A)
Of course you can also interchange the order of A and B giving:
P(B | A) = P(A | B) P(B)
Let’s take and example to illustrate this.
A bag contains 3 red balls and 1 blue ball. Two balls are selected without replacement. What is the probability that both balls are red?
Let R1 be the probability that the first ball is red
R2 be the probability that the second ball is red
So we can find P(R1) and P(R2)
Then we use the results to find P(R1 ∩ R2) as well.
P(R1) = 3/4 that is 3 red balls over total ball.
P(R2 | R1) = 2/3 since only 2 red balls is left and total balls is now 3
Using the multiplication Law, we can now find P(R1 ∩ R2)
P(R1 ∩ R2) = 3/4 x 2/3 = 1/2
This is the probability that both balls are red
Let’s take another example
Assuming that if the weather is cloudy, the probability that it is raining (A) is 0.3. Also the probability that it is cloudy is P(B) is 0.2.
What is the probability that it is cloudy and raining:
P(B) = 0.2
P(A | B) = 0.3
P(A ∩ B) = P(A | B) P(B) = 0.2 * 0.3 = 0.6
In the next lesson, we would examine Law of Total Probability