How to Find Probability of A Intersection B

Probability shows how likely an event will happen. Probability comes into application in the fields of physical sciences, commerce, biological sciences, medical sciences, weather forecasting, etc. As far as the JEE exam is concerned, probability is an important topic. The probability of an event E is defined as P(E) = [Number of favourable outcomes of E]/ [Total number of possible outcomes of E]. Events can be dependent or independent. If the probability of occurrence of one event is not affected by the occurrence of another event, then those events are known as independent events else dependent events. In this article, we will discuss independent events and how to find probability of A intersection B.

A and B are independent events if P(F/E) = P(F), P(E) not equal to zero and P(E/F) = P(E), P(F) not equal to zero.

How to Find the Probability of A ⋂ B

The probability that two events A and B both occur is the probability of the intersection of A and B. It is denoted by A⋂B.

1. When A and B are independent, the following equation gives the probability of A intersection B. P(A⋂B) = P(A).P(B)

2. When A and B are mutually exclusive events, then P(A⋂B) = 0.

3. 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 already happened.

P(A⋂B) = P(A).P(B/A)

Example

A container has 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from it. What is the probability that both of the marbles are black?

Solution:

Let A denotes the event that the first marble is black and B denotes the event that the second marble is black.

No. of black marbles = 4

Total no. of marbles = 10

P(A) = 4/10

Now we select from remaining marbles.

So P(B/A) = 3/9.

P(A⋂B) = P(A).P(B/A)

= (4/10).(3/9)

= 4/30

= 0.133