# For the following, state whether the events created are mutually exclusive and collectively...

## Question:

For the following, state whether the events created are mutually exclusive and collectively exhaustive. Respondents were classified by the type of concerts they preferred to attend: rock, jazz, classical, indoor, or outdoor. The events are neither mutually exclusive nor collectively exhaustive. The events are only mutually exclusive. The events are only collectively exhaustive. The events are both mutually exclusive and collectively exhaustive.

## Mutual exclusive and mutually exhaustive events

Mutual exclusive events are those the events under which the possibility of occurrence of any two events together is zero i.e., intersection is zero and under mutual exhaustive events the sum of probability of all possible outcomes is 1.

{eq}\begin{align*} mutually\,exclusive:P\left( {{A_i} \cap {A_j}} \right) &= 0\\ mutually\,exhaustive:P\left( {{A_i} \cup {A_j}} \right) &= 1 \end{align*} {/eq}

Where, i and j includes all possible cases.

An event is said to be mutually exclusive events when the possibility of occurrence of any two or more events together in the sample space is zero.

As

{eq}P\left( {{\rm{Concert}} \cap {\rm{outdoor}}} \right) \ne 0 {/eq}

Hence, if a person select to go to any of the 3 concert its possibility of attending the concert and outdoor will not be zero have, the events are not mutually exclusive events.

An event is said to be mutually exhaustive events when the probability of happing of all possible outcome is summed to 1 and no other events are possible.

As

{eq}P\left( {{\rm{Concert}} \cup {\rm{outdoor}}} \right) = 1 {/eq}

Thus, if a person is selected to go to any of the 3 concert he will either go to indoor or outdoor and there is no other possibility.

Hence, the event is mutually exhaustive event.

The correct option is The events are only collectively exhaustive.