Question

They gathered 200 people of which 50 were known to have the zombie virus.  of those who had the virus, 48 tested positive.  of those who did not have the virus, 135 tested negatives.

determine whether or not the events of having the virus and testing positive are independent.

Answer 1

Let
`V` be the set of people infected with the virus, and
`T` the set of people who get a positive result (true or false) when tested for the virus. Any person tested for the virus will get one or the other result and not both, so the events
`V\cap T` and
`V\cap T^C` are mutually exclusive:



`V=(V\cap T)\union(V\cap T^C)\implies |V|=|V\cap T|+|V\cap T^C|`

where
`|\cdot|` denotes the size/cardinality of a set.

We're told that
`|V|=50` and
`|V\cap T|=48`, which means
`|V\cap T^C|=50-48=2`.

Similarly, we have


`V^C=(V^C\cap T)\union(V^C\cap T^C)\implies|V^C|=|V^C\cap T|+|V^C\cap T^C|`

and we know
`|V^C|=200-50=150` and
`|V^C\cap T^C|=135`, so
`|V^C\cap T|=150-135=15`.

The probability of each possible event is obtained by dividing the size of each corresponding set by the total 200. So we have the following table:


`\begin{matrix}&V&V^C&\text{total}\\T&48&15&63\\T^C&2&135&137\\\text{total}&50&150&200\end{bmatrix}`

The events
`V` and
`T` are independent if
`P(V\cap T)=P(V)\cdot P(T)`. From the table above, we find that


`P(V\cap T)=(48)/(200)`

and


`P(V)\cdot P(T)=(50)/(200)\cdot(63)/(200)=(63)/(800)`

which are not equal, so the events are not independent.