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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
P (A cup B) = P(A) + P(B)}$
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)
= frac{4}{52} + frac{4}{52} \[7pt]
= frac{1}{13} + frac{1}{13} \[7pt]
= frac{2}{13}}$
For Non-Mutually Exclusive Events
In case there is a possibility of both events to occur then the additive theorem is written as:
P (A cup B) = P(A) + P(B) – P(AB)}$
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
= frac{3}{7}+frac{2}{5}-(frac{3}{7} times frac{2}{5}) \[7pt]
= frac{29}{35}-frac{6}{35} \[7pt]
= frac{23}{35}}$
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