A lottery has been released for sale where the total value…

A lottery has been released for sale where the total value of tickets is $n$ dinars. The entire prize amount is distributed among $m$ tickets. Find the expected value (mathematical expectation) of the prize for one ticket.

Step 1: What is given and what is missing

• total value of all tickets: $n$ dinars;
• number of winning tickets: $m$;
• price of one ticket: $r$ dinars — this is part of the original problem and is needed to find the number of tickets.

Step 2: The number of tickets

\[ N = \frac{\text{total value}}{\text{price of one ticket}} = \frac{n}{r}. \]

Step 3: The probability that one ticket wins

A bought ticket is equally likely to be any of the $N$ tickets, and $m$ of them win. By the classical definition,

\[ \boxed{p = \frac{m}{N} = \frac{m}{n/r} = \frac{mr}{n}.} \]

Step 4: The expected prize for one ticket

Suppose the total prize fund is $S$ dinars, shared by the $m$ winning tickets, so a winning ticket gets $S/m$ on average. The prize $X$ of one ticket is $0$ with probability $1-p$ and on average $S/m$ with probability $p$, so

\[ E[X] = p\cdot\frac{S}{m} + (1-p)\cdot 0 = \frac{mr}{n}\cdot\frac{S}{m} = \frac{S\,r}{n} = \frac{S}{N}. \]

In words: the expected prize of one ticket is the whole prize fund divided equally among all tickets.

• If all the money collected is paid out as prizes ($S=n$), then $E[X]=r$: on average you win back exactly the ticket price — a fair lottery.
• In real lotteries $S\lt n$, so $E[X]\lt r$: on average a ticket loses money.

Step 5: Numerical example

Tickets worth $n=100\,000$ dinars are sold at $r=20$ dinars each, so $N=5000$ tickets; $m=500$ of them win, and the prize fund is $S=40\,000$ dinars.

\[ p = \frac{500\cdot20}{100\,000} = 0.1, \qquad E[X] = \frac{40\,000}{5000} = 8 \text{ dinars}. \]

One ticket in ten wins, and on average a ticket costing $20$ dinars returns $8$ dinars.