Five hundred tickets will be sold, and these will be raffled during the town fiesta. One of these tickets will win P3,000, and the rest will win nothing. What will be the expected outcome and variance of your gain if you buy one of the tickets?

Let the random variable $Y$ be the random of the amount won. The random variable $Y$may take on only two value, (0,3000). That is $y=0,3000$ 

There are five hundred tickets to be sold and only one ticket wins P3,000. Therefore, the probability that a ticket wins P3,000 is ${1\over500}$ and the probability that a tickets wins nothing is ${1-{1\over 500}}={499\over500}$ .

We can write this as, $p(y=0)={499\over 500}$ and $p(y=3000)={1\over 500}$.

The expected value is given as, $E(y)=\sum yp(Y=y)=(0\times {499\over500})+(3000\times {1\over500})=6$ 

To find the variance, we first find $E(y^2)=\sum y^2p(Y=y)=(3000^2\times {1\over500})=18000$

Variance, $var(y)=E(y^2)-(E(y))^2=18000-36=17964$

The expected gain and the variance of your gain are 6 and 17964 respectively.