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