III. Theorem of Multiplication
When we have to determine the probability of joint occurrence or occurrence in unison of two or more than two events, the theorem of multiplication is used. For instance, the probability of getting the same number on two dice tossed simultaneously, drawing different coloured balls from a box having red, blue, and green balls.
There are 2 cases in theorem of multiplication:
1. When events are Independent
If E1 and E2 are independent events with P(E1) ≠ 0 and P(E2) ≠ 0, then according to this theorem of probability:
P(E1 ⋂ E2) = P(E1).P(E2)
This means that the probability of intersection of two events E1 and E2 is equal to the product of the individual probabilities of events E1 and E2.
If there are more than two events; say, E1, E2, and E3, then,
P(E1 ⋂ E2 ⋂ E3) = P(E1).P(E2).P(E3)
Example:
A statistics problem is given to two students, say A and B. Their chances of solving it correctly are known to be 0.5 and 0.3, respectively. Find the probability that both of them solve it.
Solution:
Let E1 be the event of A solving the problem and E2 is the event of B solving the problem. Here event E1 and E2 are independent.
Therefore, the chances of both A and B solving the problem is 15%.
2. When Events are not Independent
If the events are not independent, then multiplication theorem states that the joint probability of the events E1 and E2 is given by the probability of event E1 multiplied by the probability of event E2 given that event E1 has occurred and vice-versa. Simply put, the rule uses the concept of conditional probability when the events are known to be dependent or non-independent. According to this theorem, if E1 and E2 are two events where P(E1) ≠ 0 and P(E2) ≠ 0, and if E1 and E2 are not independent events, then:
P(E1 ⋂ E2) = P(E1).P(E2/E1)
P(E1 ⋂ E2) = P(E2).P(E1/E2)
Similarly, if there are three dependent events E1, E2, and E3, then,
P(E1 ⋂ E2 ⋂ E3) = P(E1).P(E2/E1).P(E3/E1⋂E2)
Example:
A large company employs 70 engineers, of whom 36 are males and the remaining are females. Of the female engineers. 14 are under 35 years of age, 15 are between 35 and 45 years of age, and the remaining are over 45 years of age. What is the probability of randomly selected engineer who is a female and under the age of 35 years of age?
Solution:
Let E1 represent the event that an engineer selected at random is a female and E2 is the event that a female engineer selected is under 35 years of age.
Since there are 36 males out of 70 engineers, it means that the number of female engineers is 34.
P(E1) =
P(E1) =
Therefore, P(E1⋂E2) =
P(E1⋂E2) =
P(E1⋂E2) =