Statistics - Probability Additive Theorem


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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

${P(A\ or\ B) = P(A) + P(B) \\[7pt] P (A \cup B) = P(A) + P(B)}$

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

${P(A \cup B \cup C) = P(A) + P(B) + P(C) }$

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)

${P (A \cup B) = P(A) + P(B) \\[7pt] = \frac{4}{52} + \frac{4}{52} \\[7pt] = \frac{1}{13} + \frac{1}{13} \\[7pt] = \frac{4}{13}}$

For Non-Mutually Exclusive Events

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

${P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B)\\[7pt] P (A \cup B) = P(A) + P(B) - P(AB)}$

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

${P (A \cup B) = P (A) + P(B) - P (A \cap B) \\[7pt] = \frac{3}{7}+\frac{2}{5}-(\frac{3}{7} \times \frac{2}{5}) \\[7pt] = \frac{29}{35}-\frac{6}{35} \\[7pt] = \frac{23}{35}}$


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