Conditional Probability with Example

In this class, We discuss Conditional Probability with Example.

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The reader should have prior knowledge of dependent events. Click Here.

Conditional Probability

P(A|B) = P(A ∩ B) / P(B)

P(A|B) = probability of event A given event B happened.

We understand the conditional probability equation with an example.

Example:

Random Experiment: Toss two dice.

Event A = the sum of the values on the two dice is seven.

Sample space S = 36

A = {(5,2), (2,5), (3,4), (4,3), (6,1),(1,6}

P(A) = 6/36

If there is no condition our sample space S = 36.

We are using the full sample space.

Conditional probability P(A|B)

Event B = obtained five on one of the dice.

Event A = sum of the values on the dice = 7

It was given event B happened whenever event B happened sample space changes.

The new sample space S1 = {(1,5), (5,1), (2,5), (5,2), (3,5), (5,3), (5,4), (4,5), (5,5), (6,5), (5,6)}

S1 = 11

the happening event A called conditional probability in the new sample space.

There are two chances in the new sample space {(2,5), (5,2)}

P(A|B) = 2/11

Now we relate the conditional probability value 2/11 with our equation.

P(A|B) = P(A ∩ B) / P(B)

P(A ∩ B) = 2/36

P(B) = 11/36

If we cancel 36 from the numerator and denominator, we get 2/11