According to this rule, probabilities of two or more related events are multiplied with each other to find out the net probability of joint occurrence. If A and B are any two events associated with a random experiment, then multiplication rule for A and B is given as:

**P(A and B) = P(A). = P(B).**

**Special case of the multiplication rule:**

If A and B are any two **independent events**, then multiplication rule for A and B is given as:**P(A and B) = P(A ∩ B) = P(A).P(B)**

Ex: |
An unbiased die is thrown. If A is the event that 'the number appearing is prime' and B be the event that 'the number appearing is more than 4', then find whether A and B are independent ? |

Sol: |
We know that the sample space is: S = {1, 2, 3, 4, 5, 6} Now, A = {2, 3, 5}, B = {5, 6} and A ∩ B = {5}. Then, P(A) = = , P(B) = = and P(A ∩ B) = Evidently, P(A ∩ B) = P(A).P(B) Hence, A and B are independent events. |

Ex: |
A bowl contains ten chips. Four of them are red and remaining blue. If two chips are picked with out replacement, find the probability of the first chip being red and the second blue. |

Sol: |
There are 10 chips in all in the bowl. Let A be the event of picking a red chip and B that of picking blue chip. So P(A) = 4/10 and P(B/A) = 6/9 P(A ∩ B) = .P(A) = (6/9).(4/10) = 4/15 |

## Independent events:

Two or more events are said to be **independent** if occurrence or non-occurrence of any of them does not affect the occurrence or non-occurrence of the other event. For example, when a die is rolled twice, the event of occurrence of '1' in the first throw and the event of occurrence of '1' in the second throw are independent events.

If A and B are any two independent events, then

P(A) = and P(B) =

## Dependent events:

Two events are said to be **dependent** if the outcome (or occurrence) of the first effects the outcome(or occurrence) of the second.

## 3 mutually independent events

**Note: ** Three events A, B, C are said to be mutually independent, if

P(A ∩ B) | = | P(A).P(B) |

P(B ∩ C) | = | P(B).P(C) |

P(A ∩ C) | = | P(A).P(C) and |

P(A ∩ B ∩ C) | = | P(A).P(B).P(C) |

If at least one of the above is not true for three given events, we say that the events are not independent.

Some conditional probability problems can be solved by using a tree diagram. A tree diagram is a schematic way of looking at all possible outcomes.

For the case of dependent events, we have seen how to calculate the probability of occurrence of event B if we know that event 'A' has already occurred . Can we calculate the probability of occurrence of 'A' if we know for sure that the event 'B' has occurred? We indicate such probability by P(A/B).

The occurrence of event B happens through two paths as shown below.

The above is explained through an example at the adjacent.