# 8. Independent and Dependent Events

If the occurrence or non-occurrence of *E*_{1} does not affect the probability of occurrence of *E*_{2}, then

P(E_{2}|E_{1}) =P(E_{2})

and *E*_{1} and *E*_{2} are said to be **independent events**.

Otherwise they are said to be **dependent** events.

[Recall from Conditional Probability that the notation *P*(*E*_{2} | *E*_{1}) means "the probability of the event *E*_{2} given that *E*_{1} has already occurred".]

## Two Events

Let's consider "*E*_{1} and *E*_{2}" as the event that "both *E*_{1} and *E*_{2} occur".

If *E*_{1} and *E*_{2} are **dependent events**, then:

P(E_{1}andE_{2}) =P(E_{1}) ×P(E_{2}|E_{1})

If *E*_{1} and *E*_{2} are **independent events**, then:

P(E_{1}andE_{2}) =P(E_{1}) ×P(E_{2})

## Three Events

For three **dependent **events *E*_{1}, *E*_{2}, *E*_{3}, we have

P(E_{1}andE_{2}andE_{3}) =P(E_{1}) ×P(E_{2}|E_{1}) ×P(E_{3}|E_{1}andE_{2})

For three **independent **events *E*_{1}, *E*_{2}, *E*_{3}, we have

P(E_{1}andE_{2}andE_{3}) =P(E_{1}) ×P(E_{2}) ×P(E_{3})

### Example 1

If the probability that person *A* will be alive in `20` years is `0.7` and the probability that person *B* will be alive in `20` years is `0.5`, what is the probability that they will both be alive in `20` years?

### Example 2

A fair die is tossed twice. Find the probability of getting a `4` or `5` on the first toss and a `1`, `2`, or `3` in the second toss.

### Example 3

Two balls are drawn successively without replacement from a box which contains `4` white balls and `3` red balls. Find the probability that

(a) the first ball drawn is white and the second is red;

(b) both balls are red.

### Example 4

A bag contains `5` white marbles, `3` black marbles and `2` green marbles. In each draw, a marble is drawn from the bag and not replaced. In three draws, find the probability of obtaining white, black and green in that order.

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