From Prize Amounts to Expected Net Gain
In Interpreting Expected Value as a Long-Run Average, you learned that the expected value of a random variable is the probability-weighted average of its possible values. In a raffle, the values of interest are often not the prizes themselves, but the ticket buyer’s net gain: the prize value minus the price paid for the ticket.
A ticket can have several possible outcomes: it might win the top prize, win one of several smaller prizes, or win nothing. To calculate expected net gain, account for every outcome and its probability, including the no-prize outcome. The result is an average per ticket under the raffle’s probability model; it does not predict what any particular ticket will win.
This builds on Probability Distributions for Games of Chance, where you represented a ticket’s result with a random variable and included the ticket cost. Here, the key task is using the probability of each prize level to find the expected value when there are several possible prizes.
Assume the raffle rules assign each ticket at most one prize, so winning different prize levels and winning nothing are mutually exclusive outcomes. When tickets are equally likely to be drawn, the probability of a prize level is the number of tickets that win that prize divided by the total number of tickets.
A Reliable Calculation Process
For each possible outcome, first determine the prize value and subtract the ticket cost. Then multiply that net gain by the outcome’s probability. Add the products to find \(\mu_X\). This follows the mean formula from The Mean of a Discrete Random Variable, applied to net gains instead of prize amounts.
A ticket that wins nothing still has a net gain: \(0-c=-c\). Omitting that outcome or treating it as a net gain of zero would leave out the cost paid for losing tickets and make the expected value too high.
Include every prize level and the outcome of winning no prize.
For equally likely tickets, divide the number of tickets in an outcome category by the total number of tickets.
Subtract the ticket cost from each prize value, including a prize value of zero for no prize.
Multiply each net gain by its probability and add the products. Check that the probabilities sum to 1.
There is also a useful check when every ticket costs the same amount. Calculate the expected prize value first, then subtract the fixed ticket cost. This gives the same answer because the cost is paid on every outcome.
The last equality uses the fact that the probabilities of all possible outcomes sum to 1. This shortcut is a check on the net-gain calculation; listing the net gains directly makes it easier to see that losing tickets and ticket costs have been handled correctly.
Worked Example: A Raffle with Three Prize Levels
Worked Example: A Raffle with Three Prize Levels
An invented community raffle sells 500 tickets for $2 each. One ticket wins $200, four tickets win $25 each, and 20 tickets win $5 each. Every other ticket wins no prize. Let \(X\) be the net gain from buying one ticket. Calculate and interpret \(\mu_X\).
Find the probabilities. There are \(500-1-4-20=475\) no-prize tickets. Since each ticket is equally likely to be drawn, the probabilities are \(1/500=0.002\) for the $200 prize, \(4/500=0.008\) for a $25 prize, \(20/500=0.040\) for a $5 prize, and \(475/500=0.950\) for no prize. They sum to \(0.002+0.008+0.040+0.950=1.000\).
Find the net gain for each outcome. Subtract the $2 cost from each prize value. The possible net gains are \(200-2=\$198\), \(25-2=\$23\), \(5-2=\$3\), and \(0-2=-\$2\) for no prize.
| Outcome | Net gain, \(x\) | \(P(X=x)\) | \(xP(X=x)\) |
|---|---|---|---|
| $200 prize | $198 | 0.002 | \(198(0.002)=0.396\) |
| $25 prize | $23 | 0.008 | \(23(0.008)=0.184\) |
| $5 prize | $3 | 0.040 | \(3(0.040)=0.120\) |
| No prize | -\$2 | 0.950 | \((-2)(0.950)=-1.900\) |
Calculate the expected net gain. Multiply each possible net gain by its probability and add:
Check using expected prize value. The expected prize value is \(200(0.002)+25(0.008)+5(0.040)+0(0.950)=0.400+0.200+0.200=\$0.800\) per ticket. Subtracting the $2 ticket cost gives \(\$0.800-\$2.000=-\$1.200\) per ticket, matching the direct calculation.
Interpret in context. The expected net gain is \(-\$1.20\) per ticket. Under this probability model, the average net gain across many independent repetitions under the same conditions tends toward a loss of about $1.20 per ticket. A particular ticket can still win a prize, so this average does not say what one ticket will receive.
Why the Losing Tickets Matter
In a raffle, most tickets may win nothing. Their net gains are negative because each ticket has a cost. The number and probability of losing tickets therefore have a major effect on expected net gain. A large top prize can attract attention, but its probability must be included along with the probabilities of every smaller prize and of winning nothing.
Also keep the probability denominator straight. If 30 tickets win a particular prize in a raffle with 800 total tickets, the probability of that prize on one ticket is \(30/800\), not \(30\) divided by the number of winning tickets. The ticket buyer could draw any of the 800 tickets.
Worked Example: Include Every Prize and the No-Prize Outcome
An invented school raffle sells 800 tickets at $5 each. One ticket wins $500, five tickets win $40 each, and 30 tickets win $10 each. The other tickets win nothing. Find the expected net gain per ticket using the direct net-gain method, then check it using expected prize value.
Find the number and probability of no-prize tickets. The number is \(800-1-5-30=764\). The probabilities for the $500, $40, $10, and no-prize outcomes are \(1/800=0.00125\), \(5/800=0.00625\), \(30/800=0.03750\), and \(764/800=0.95500\). These add to \(1.00000\).
Convert each outcome to net gain. After subtracting the $5 ticket cost, the net gains are $495, $35, $5, and \(-\$5\), respectively.
Check with expected prize value. The expected prize value is \(500(0.00125)+40(0.00625)+10(0.03750)+0(0.95500)=0.625+0.250+0.375=\$1.25\). Subtracting the $5 cost gives \(\$1.25-\$5.00=-\$3.75\) per ticket, the same result.
Thus, the expected net gain is \(-\$3.75\) per ticket. It describes the probability-weighted average after accounting for both prizes and the ticket cost. It does not mean that a ticket buyer will literally receive a loss of exactly $3.75 on one ticket; the individual net gain must be one of the listed possible values.
Worked Example: A Smaller Raffle with Several Winners
Worked Example: A Smaller Raffle with Several Winners
An invented neighborhood raffle has 250 tickets costing $3 each. One ticket wins $250, five tickets win $20 each, and ten tickets win $5 each. All remaining tickets win nothing. Calculate the expected net gain and explain the role of the multiple smaller prizes.
Determine the probabilities. The number of no-prize tickets is \(250-1-5-10=234\). The probabilities are \(1/250=0.004\), \(5/250=0.020\), \(10/250=0.040\), and \(234/250=0.936\). Their sum is \(0.004+0.020+0.040+0.936=1.000\).
Find the net gains. The net gains are \(250-3=\$247\), \(20-3=\$17\), \(5-3=\$2\), and \(0-3=-\$3\) for no prize.
Verify using expected prize value. The expected prize value is \(250(0.004)+20(0.020)+5(0.040)=1.000+0.400+0.200=\$1.600\) per ticket. Subtract the $3 cost: \(\$1.600-\$3.000=-\$1.400\), which agrees with the direct calculation.
The five $20 prizes and ten $5 prizes are separate prize levels, each with its own probability and net gain. Their contributions to the expected prize value are $0.40 and $0.20 per ticket, respectively. A prize level with many winners can contribute meaningfully even when each individual prize is modest; the expected value accounts for both the amount and the probability.
Common Mistakes and AP Exam Tips
- Using prize values instead of net gains without subtracting the ticket cost. Prize values alone give the expected prize value, not the ticket buyer’s expected net gain. Either subtract the cost from every outcome or subtract the fixed cost once after finding expected prize value.
- Leaving out the no-prize outcome. Include its probability and use a net gain of \(-c\). A losing ticket still costs money.
- Dividing by the number of prizes instead of the number of tickets. When every ticket is equally likely to be drawn, divide the number of tickets in an outcome category by the total number of tickets.
- Adding prize amounts without their probabilities. A rare top prize and a frequently awarded smaller prize do not contribute equally just because both are listed. Use \(xP(X=x)\) for every outcome.
- Misreading a negative expected value as a guaranteed loss on one ticket. A negative expected net gain is an average across the probability model, not a prediction that each ticket loses that exact amount.
- Reporting only a number without context or units. A full-credit interpretation identifies the quantity as an average net gain per ticket and states the long-run meaning. For example: “The expected net gain is \(-\$1.40\) per ticket, so across many independent repetitions under the same conditions, the average net gain tends toward a loss of about $1.40 per ticket.”
Before finalizing a calculation, check that every outcome is included, the probabilities add to 1, and each net gain equals prize value minus ticket cost. Then compare the direct weighted sum with expected prize value minus cost. Agreement between the two calculations is a useful arithmetic check.
Key Takeaway
Expected net gain brings prize chances and ticket cost together in one probability-weighted average. The value can be positive, zero, or negative, and it summarizes the chance model rather than promising a particular ticket result.
Check Your Understanding
Use this invented raffle: 400 equally likely tickets cost $4 each. One ticket wins $300, three tickets win $30 each, and 16 tickets win $5 each. The remaining tickets win nothing.
- How many tickets win no prize, and what is the probability of winning no prize?
- What is the net gain for each of the four outcomes: the $300 prize, a $30 prize, a $5 prize, and no prize?
- Calculate the expected net gain per ticket by showing each probability-weighted net gain and adding the products.
- Calculate the expected prize value and use it to check your answer to Question 3.
- Write a context sentence interpreting the expected net gain. Explain why it does not predict the result of one particular ticket.