Nikbakht has received an email from an unknown mailer offering an entry into a lottery. For a price, he can get a ticket that may pay a prize double the price. He buys 2 tickets. Unknown to him an unfair system determines a winner where losing is two times more likely than winning. In what ways is this lottery unfair? What is the expected value of this lottery? In your estimation how much has he paid?

Solution:

Let the amount he pays for 1 ticket be $x$.

Paid price for 2 tickets $=2x$

Winning amount $=2(2x)=4x$

Let the probability of winning$=p$ and probability of losing $=q$

$\therefore p=\dfrac 13, q=\dfrac 23$

Lottery is unfair as probability of losing is more than winning, i.e., $q>p$

$E[X]=4x\times \dfrac13-2x\times \dfrac23=\dfrac{4x}3-\dfrac{4x}3=0$

So, expected value is 0.

He has paid $2x$ and expected value is 0.

So, estimation is that whatever the amount he has paid ($2x$ in our case), for buying 2 tickets, is all in vain.