What is Independent Events in Probability?

What is Independent Events in Probability?

The definition of independence in probability: Event A and B are independent if and only if P(A ∩ B) = P(A) x P(B). Sometimes people misuse the terms independence of events with mutually exclusive or disjoint events or vice versa. Don’t get confused with those two term  independence of events is completely different property. To get clear view visit this article.

If A and B are independent then

P( A | B ) = [ P(A) x P(B) ] / P(B) = P(A) for P(B) > 0

P( B | A) = [ P(A) x P(B)] / P(A) = P(B) for P(A) > 0

Check for Independence of Two Events
When A and B have non – zero probabilities of occurring; below 3 statements are equivalent:
1. P( A ∩ B ) = P(A) x P(B)
2. P( A | B ) = P(A)
3. P( B | A ) = P(B)

If they are true, then A and B are independent.
If they are false, then A and B are non independent.

Example 1:
Suppose P(A) = 0.7 and P(B) = 0.2 and A and B are independent. What is P( A ∩ B )?
solution:
P( A ∩ B ) =P(A) x P(B) = 0.7 x 0.2 = 0.14
Here, Since A and B are independent. Conditional Probability of:
P (A | B) = 0.7 and P(B | A) = 0.2

Example 2:
Suppose P(A) = 0.4 and P(B) = 0.8, and  P( A ∩ B ) = 0.36. Are A and B independent?
solution:
we have P( A ∩ B ) = 0.36
P(A) x P(B) = 0.4 x 0.8 = 0.32
P(A) x P(B) ≠ P( A ∩ B )
Hence, A and B are not independent.
P( A | B ) = P( A ∩ B ) / P(B) = 0.36 / 0.8 = 0.45 ≠ P(A)
P( B | A ) = P( B ∩  A) / P(A) = 0.36 / 0.4 = 0.9 ≠ P(B)

NOTE:
When two events are independent, the occurrence of one event does not change the probability of the other event.