Question

A group of people were asked if they enjoy going to concerts or movies.

The table shows the probabilities of the results.

Answer 1

The correct answer is the second choice.

They are not independent because P(movies|concerts)=0.25 ≠ P(movies)=0.6, and P(concerts|movies)=0.25 ≠ P(concerts)=0.55

Answer 2

Option: B is the correct answer.

Enjoying movies and concerts are not independent since,

`P(movies|concerts)\\eq P(movies)`

and
`P(concerts|movies)\\eq P(concerts)`

Explanation:

We know that if two events A and B are independent then:

`P(A\bigcap B)=P(A)* P(B)`

where P denote the probability of an event.

Now, we know that:

`P(movies|concerts)=(P(movies\bigcap Concerts))/(P(concerts))`

Similarly:

`P(concerts|movies)=(P(movies\bigcap Concerts))/(P(movies))`

If movies and concerts are independent then:

`P(movies|concerts)=P(movies)`

Similarly:

`P(concerts|movies)=P(concerts)`

We have:

`P(movies)=0.6`

`P(concerts)=0.55`

`P(concerts|movies)=(0.25)/(0.6)`

`P(movies|concerts)=(0.25)/(0.55)`

Hence,

Enjoying movies and concerts are not independent

Since,

`P(movies|concerts)\\eq P(movies)`

and
`P(concerts|movies)\\eq P(concerts)`