# Consider the following statement: Let A,B,C be three events. If P(A) + P(B) + P(C) = 1 then the...

## Question:

Consider the following statement: Let A,B,C be three events. If P(A) + P(B) + P(C) = 1 then the events A,B,C are mutually exclusive.

Is this statement true or false? If true, prove it and if false find a counterexample.

## mutually exclusive events:

The two events A and B are said to be mutually exclusive events if their intersection is zero. A simple example is the tossing of a coin in which there can be either tail or head but not both.

Given:

A, B and C are the three events

If {eq}P\left( A \right) + P\left( B \right) + P\left( C \right) = 1 {/eq} then the events are said to be mutually exclusive.

The above statement is true.

In case of mutually exclusive events

{eq}\begin{align*} P\left( {A \cap B} \right) &= 0\\ P\left( {B \cap C} \right)& = 0\\ P\left( {C \cap A} \right) &= 0\\ P\left( {A \cap B \cap C} \right)& = 0 \end{align*} {/eq}

Using

{eq}\begin{align*} P\left( {A \cup B \cup C} \right) &= P\left( A \right) + P\left( B \right) + P\left( C \right) - P\left( {A \cap B} \right) - P\left( {B \cap C} \right) - P\left( {C \cap A} \right) + P\left( {A \cap B \cap C} \right)\\ & = P\left( A \right) + P\left( B \right) + P\left( C \right) - 0 - 0 - 0 + 0 \le 1\\ \Rightarrow P\left( A \right) + P\left( B \right) + P\left( C \right) &= 1 \end{align*} {/eq}