Statistics - Probabipty Additive Theorem

For Mutually Exclusive Events

The additive theorem of probabipty states if A and B are two mutually exclusive events then the probabipty 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 probabipty 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{2}{13}}$

For Non-Mutually Exclusive Events

In case there is a possibipty 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 probabipty of the target being hit at all when both of them try.

Solution:

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

Probabipty 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 apppcable is

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

Statistics - Probabipty Additive Theorem

For Mutually Exclusive Events

Example

For Non-Mutually Exclusive Events

Example

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