B. Probability Distributions · Discrete Probability Distributions

Expected Value in Games, Raffles and Lotteries

B1.2 B1.7About 70 minutes

Learning goals

I can find the expected net gain of a game by including the cost to play.
I can decide whether a game is fair and find the price or prize that would make it fair.
I can use expected value to analyse raffles, lotteries and casino games, and explain why the organizer profits.
I can explain why expected value describes the long run, and why people may still choose differently in a single decision.

Words to know

expected net gain
The expected value of (winnings − cost to play). Also called expected payoff or expected profit.

fair game
A game whose expected net gain is 0 for each player: nobody wins or loses on average.

house edge
The organizer's average profit per dollar bet; the negative of the player's expected net gain per dollar.

fair price
The cost to play that makes a game fair: it equals the expected value of the winnings.

Should you play?

At a school fun fair, a booth says: "$2 to play! Roll a 6 and win $6!" That sounds like a 3-to-1 payoff. Is it a good deal?

The key is to look at your net gain: what you win minus what you paid.

Roll a 6 (probability 1/6): you get $6 back after paying $2, so your net gain is +$4.
Any other roll (probability 5/6): your net gain is −$2.

E(net) = 4(1/6) + (−2)(5/6) = 4/6 − 10/6 = −$1.

On average you lose $1 every time you play. Over 100 plays you would expect to be down about $100, and the booth would expect to be up about $100. That money is the booth's profit.

Fair games

A game is fair if the expected net gain is 0. Two equivalent ways to check:

E(net gain) = 0, or
E(winnings) = cost to play.

The second way gives the fair price: the fair price of a game equals the expected value of what you win. For the die booth, E(winnings) = 6(1/6) = $1, so the fair price is $1, not $2.

Games run to raise money (raffles, fun fairs) and commercial games (lotteries, casinos) are deliberately not fair. The organizer's average profit per dollar bet is called the house edge.

Worked example

Example 1: Making a game fair

You pay $1 to roll two dice. If the sum is 2 or 12, you collect your $1 back plus a net prize of $k. Otherwise you lose your $1. Find k so the game is fair.

P(2 or 12) = 2/36. P(other) = 34/36.
E(net) = k(2/36) + (−1)(34/36).
For a fair game, set E(net) = 0: 2k/36 = 34/36, so 2k = 34 and k = 17.

A net prize of $17 (so you collect $18 in total) makes the game fair. Any smaller prize favours the organizer.

A two-stage game drawn as a tree. Multiply along each branch to get the probability of each final net gain, then use E = Σ x·P(x).

Worked example

Example 2: A two-stage game

Pay $3. Toss a coin. On heads, spin a spinner with 4 equal sectors: one sector wins $10, the other three win nothing. On tails, win $2. Find the expected net gain and the fair price.

Final outcomes (multiply along branches): heads then $10: (1/2)(1/4) = 1/8, net 10 − 3 = +7.
Heads then $0: (1/2)(3/4) = 3/8, net −3. Tails: 1/2 = 4/8, net 2 − 3 = −1.
Check: 1/8 + 3/8 + 4/8 = 1. ✓
E(net) = 7(1/8) + (−3)(3/8) + (−1)(4/8) = (7 − 9 − 4)/8 = −6/8 = −0.75.
E(winnings) = 10(1/8) + 0 + 2(4/8) = 1.25 + 1 = 2.25.

Expected net gain −$0.75 per game. The fair price is $2.25.

Worked example

Example 3: A charity raffle

A hospital foundation sells 2000 raffle tickets at $5 each. Prizes: one $1000 gift card, two $250 gift cards and five $50 gift cards. You buy one ticket. Find your expected net gain and the foundation's expected profit.

Total prize value: 1000 + 2(250) + 5(50) = 1750.
E(winnings) for one ticket = 1000(1/2000) + 250(2/2000) + 50(5/2000) = 1750/2000 = $0.875.
E(net) = 0.875 − 5 = −$4.125.
Foundation: takes in 2000 × 5 = $10 000 and pays $1750 in prizes (if all tickets sell). Profit $8250.

Your expected net gain is about −$4.13 per ticket. The foundation expects $8250 profit. People buy charity tickets mostly to support the cause; expected value shows the real cost of that support.

Lotteries and casino games

The same calculation works for every game of chance.

Roulette (wheel with 38 pockets: 18 red, 18 black, 2 green). A $1 bet on red wins $1 (net) with probability 18/38 and loses $1 with probability 20/38. E(net) = 18/38 − 20/38 = −2/38 ≈ −$0.053. That is about 5.3 cents lost per dollar, on every spin, forever.
A "pick 3" lottery: choose a number from 000 to 999 for $1. If your number is drawn you win $500. E(winnings) = 500(1/1000) = $0.50, so E(net) = −$0.50 per dollar.
Lotto 6/49: to win the classic jackpot you must match all 6 numbers drawn from 49. The number of possible tickets is C(49, 6) = 13 983 816 (from Permutations and Combinations), so P(jackpot) = 1/13 983 816, less than 1 in 13 million.

Expected net gain per $1 for the games in this lesson. Every bar is negative: in each game the organizer expects to keep part of every dollar. A fair game would have no bar at all.

Know the rules, and know where to get help

Ontario sets minimum ages for gambling: 19 to enter a casino or to play on a regulated online gaming site (where 18-year-olds may only buy lottery tickets), and no lottery product may be sold to anyone under 18. Games of chance are designed to have a negative expected value for players. If gambling ever stops feeling like entertainment for you or someone you know, Ontario has free, confidential help lines and counselling, and a school counsellor or doctor can connect you with them.

Worked example

Example 4: Insurance and extended warranties

A phone store offers a 1-year protection plan for $96. Suppose that in a year, 15% of phones need a repair that would cost $400 without the plan, and the rest need nothing. Should you buy the plan, based on expected value? Why might someone buy it anyway?

Without the plan, your cost C is $400 with probability 0.15 and $0 with probability 0.85.
E(C) = 400(0.15) + 0(0.85) = $60.
With the plan, the cost is a sure $96.
The plan costs $96 − $60 = $36 more than the expected repair cost. That $36 per customer is the store's expected profit on the plan (before its own costs).

On average, skip the plan (expected cost $60 < $96). But someone who could not afford a sudden $400 bill might choose the plan to avoid that risk. Expected value gives the long-run cost; it does not measure how much a single bad outcome would hurt.

Just for fun

Riddle: what is the most reliable way to make money at a casino?

Own one. Every expected-value calculation in this lesson agrees.

Hands-on activity

Design a fair fun-fair game

You need: two dice (or a deck of cards), paper, pencil and a calculator.

Invent a game that costs $2 to play (pretend money or counters only) and pays a prize for some outcome, for example "win if the dice show doubles" or "win if you draw a heart".
Find the probability of winning. Then choose the prize so the game is fair (expected net gain 0).
Change the prize so the booth makes an expected profit of $0.50 per play. (This is what a school fundraiser would do.)
Play your game 30 times with a family member as the customer and keep track of the booth's total profit. Compare it with 30 × $0.50 = $15.

What to look for: for doubles, P = 6/36 = 1/6; a fair game pays a total of $12 to the winner (net $10). For a $0.50 expected profit, E(winnings) must be $1.50, so the total prize is $9 (net $7). Over only 30 plays, the actual profit could easily be well above or below $15.

Your turn

A game costs $5. You draw one card from a standard deck; an ace pays $26 (total), anything else pays nothing. Find the expected net gain.

What is the fair price of the card game in the previous question?

Spin a spinner with 5 equal sectors. One sector pays $20, two pay $5, two pay $0. What is the fair price to spin?

A coin game: pay $1, toss two coins. Two heads pays $3 (total), anything else pays nothing. Is it fair?

1000 raffle tickets are sold at $2. There is one prize of $500. Find the expected net gain of one ticket.

In European roulette there are 37 pockets (18 red, 18 black, 1 green). Find the expected net gain of a $1 bet on red.

Explain, in one sentence, what an expected net gain of −$0.75 means.

More practice: warm-up

What is the expected net gain in a fair game?

A game pays $10 with probability 0.1. What is E(winnings)?

If E(winnings) is $3 and it costs $4 to play, what is E(net)?

What is P(winning the Lotto 6/49 jackpot with one ticket)?

Who profits in a game with negative expected net gain for players?

More practice: on your own

Pay $2 and roll two dice. Doubles pay $8 (total); a sum of 7 pays $4 (total); anything else pays nothing. Find E(net).

A game: pay $3, roll a die and win the number shown in dollars. Find E(net) and the fair price.

A carnival game has a 1/20 chance of winning a prize worth $15 and costs $1. Find the expected profit for the carnival over 400 plays.

A lottery sells 500 000 tickets at $3 and pays one prize of $250 000 and 100 prizes of $1000. Find the house edge per dollar.

A $40 bicycle repair warranty covers a $200 repair. What probability of needing the repair would make the warranty a fair deal?

Challenge

A game: pay $1 and choose a number from 1 to 6. Roll three dice. If your number appears on exactly k dice (k = 1, 2, 3) you get back your $1 plus $k. If it does not appear, you lose your $1. Find E(net). (Hint: the number of matches follows the pattern of 3 independent trials with probability 1/6.)

A raffle sells n tickets at $10 and gives one prize of $2000. For which values of n does a ticket have a positive expected net gain? For which n is it fair?

Two games. Game A: win $100 with probability 0.5, else $0. Game B: win $50 for sure. Both cost $45. Compare their expected net gains, and explain what expected value does not tell you about these games.

Check your understanding

Quiz

A game costs $4. You win $10 (total) with probability 0.3; otherwise you win nothing. What is the expected net gain?

A game is fair when...

A spinner pays $12 with probability 1/4 and nothing otherwise. What is the fair price to play?

In roulette with 38 pockets (18 red), what is the expected net gain of a $1 bet on red?

2000 raffle tickets at $5 each; total prizes worth $1750. What is the expected net gain of one ticket?

Why might a person buy insurance even though its expected value is negative for them?

For parents and teachers

Big idea (B1.2, B1.7): the expected net gain of a game is E(winnings) − cost. A fair game has expected net gain 0. Raffles, lotteries and casino games are built to have negative expected value for players, which is how they raise money or make a profit.

A note for families: this lesson uses gambling as a context because the curriculum asks students to analyse games of chance and lotteries. It is a good chance to talk about why "the house always wins" and about Ontario's minimum ages (19 for casinos and regulated online gaming sites, and 18 for any lottery product, including lottery tickets bought online).

How to help: pick a game you see (a raffle at a hockey game, a fair booth) and estimate its expected value together.

Common difficulties:

Forgetting to subtract the cost to play, or subtracting it twice ("win $6" may already include the stake).
Thinking a negative expected value means you will lose every time.
Confusing "fair" (expected net 0) with "equally likely to win or lose".