Expected Payoff of a Game

Lesson 1 of 2: Net Payoff and Simple Games

In this lesson:

• Build a signed net-payoff table
• Compute the expected payoff per play
• Classify a game as fair, favorable, or unfavorable

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Build a signed net-payoff table, subtracting the cost to play
2. Compute the expected payoff per play of a game
3. Classify a game as favorable, fair, or unfavorable to the player

A $100 Prize for a $2 Ticket?

A booth offers a $100 prize for a $2 ticket.

Over many plays, do you come out ahead or behind?

The Payoff Is the Net: Prize Minus Cost

The payoff of a play is the net: what you receive minus the cost to play.

The Cost Comes Out of Every Outcome

• You pay the cost whether you win or lose
• So subtract it on every row of the table
• A clean rule: each row's payoff = prize − cost

Worked Example: The $2 Ticket Game

$2 ticket, $100 prize:

• Win:
• Lose:

The $2 is subtracted whether you win or lose.

Your Turn: Build a Signed Table

A $5 raffle ticket: win a $50 prize or win nothing.

Build the signed net-payoff table (net value for each outcome).

Fill in both rows before advancing.

You Have the Table: Now Weight and Sum

The signed table lists each net payoff and its probability.

The expected payoff is just the expected value of that table.

Expected Payoff Is a Weighted Sum

• The average gain or loss per play
• Same engine as — now the values are net payoffs

A Carnival Spinner Game to Score

Pay $1 to spin. Land on $0 (prob ), \tfrac38$), or \tfrac18$).

Compute: This Spinner Is Fair

So the expected payoff is 0 — a fair game.

Classify the Game by Its Sign

• Favorable to the player: expected payoff
• Fair: expected payoff
• Unfavorable: expected payoff (favors the operator)

Shift the Odds, Flip the Sign

Make the $5 less likely: probability instead of .

Now the expected payoff is negative — unfavorable.

Your Turn: Build, Compute, Classify

A $3 game: win $10 with probability , else nothing.

Build the signed table, compute the expected payoff, and classify it.

Do all three before advancing.

Two Payoff Traps to Avoid

Subtract the cost on every row — even the losing one is

A "fair" game needs the cost included — only after subtracting it

What You Built In This Lesson

✓ Net payoff = prize − cost, on every row

✓ Expected payoff = weighted sum of net payoffs

✓ Its sign classifies the game

Coming Up Next: The State Lottery

A lottery has many outcomes, a tiny win chance, and a huge prize.

Next lesson: the same method at scale — and why a million-dollar jackpot barely moves the expected payoff.