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# Statistics - Probability Additive Theorem

## For Mutually Exclusive Events

The additive theorem of probability states if A and B are two mutually exclusive events then the probability of either A or B is given by

The theorem can he extended to three mutually exclusive events also as

### Example

**Problem Statement:**

A card is drawn from a pack of 52, what is the probability that it is a king or a queen?

**Solution:**

Let Event (A) = Draw of a card of king

Event (B) Draw of a card of queen

P (card draw is king or queen) = P (card is king) + P (card is queen)

## For Non-Mutually Exclusive Events

In case there is a possibility of both events to occur then the additive theorem is written as:

### Example

**Problem Statement:**

A shooter is known to hit a target 3 out of 7 shots; whet another shooter is known to hit the target 2 out of 5 shots. Find the probability of the target being hit at all when both of them try.

**Solution:**

Probability of first shooter hitting the target P (A) = ${\frac{3}{7}}$

Probability of second shooter hitting the target P (B) = ${\frac{2}{5}}$

Event A and B are not mutually exclusive as both the shooters may hit target. Hence the additive rule applicable is