Pricing a Carnival Game and a Grand-Prize Proposal
The senior class is designing a fundraising game for the spring carnival. A player pays $4 to reach into a bag of 30 tokens and draw one; the token is returned to the bag after every play, so the bag is always the same. Three tokens are gold and win a $10 prize, seven are silver and win a $3 prize, and the remaining 20 are blank. Prizes are paid in cash from the till at the moment of the win. The class has already spent $60 on booth materials that will not be recovered, and the advisor wants the booth to net at least $1,200 over the day. One officer proposes adding a single $50 "grand prize" token to the bag, replacing one blank and bringing the bag to 31 tokens, arguing that "a big prize is what pulls a crowd." Another officer worries that the change will quietly wreck the numbers.
1. Compute the expected net gain per play for the original game, and the expected profit over 400 plays after the materials cost.
Read the worked answer
Expected payout per play:
(3/30)($10) + (7/30)($3) + (20/30)($0) = $1.00 + $0.70 + $0 = $1.70
Expected net gain to the class per play = $4.00 − $1.70 = $2.30.
Over 400 plays: 400 × $2.30 = $920 gross, minus the $60 already spent on materials = $860 expected profit.
This is a long-run average, not a guarantee. Payout per play has a standard deviation of about $3.03, so over 400 plays the total payout varies by roughly $61 - the $340 shortfall against the goal is far too large to be explained by bad luck.
2. How many plays are needed to reach the $1,200 net-profit target with the original game?
Read the worked answer
Set up the inequality with the materials cost included:
2.30n − 60 ≥ 1,200 2.30n ≥ 1,260 n ≥ 547.8
So 548 plays are needed, since plays are whole numbers and 547 leaves the class just under target.
At 400 plays the booth is $340 short, about 148 plays' worth of expected profit. Before committing to the game, the class should estimate whether a carnival crowd will supply about 550 plays; if not, the price or the prize structure has to change, not the target.
3. Evaluate the grand-prize proposal. Recompute the expected value and the plays needed, and state what the class should do.
Read the worked answer
The new bag has 31 tokens: 1 grand prize ($50), 3 gold ($10), 7 silver ($3), 20 blanks.
Expected payout = (1/31)($50) + (3/31)($10) + (7/31)($3) = ($50 + $30 + $21)/31 = 101/31 ≈ $3.26
Expected net per play = $4.00 − $3.26 = $0.74, down from $2.30 - the game now keeps about a third as much per play.
Plays to reach $1,200: (1,200 + 60) ÷ 0.742 ≈ 1,699, versus 548 before. That is roughly 3.1 times the traffic for the same result, which the booth almost certainly will not see.
The proposal is not automatically wrong, but it must be paired with a price change: at $4.75 per play the net per play is $4.75 − $3.26 = $1.49, requiring about 846 plays. A better option is to keep the 30-token bag and make the grand prize a donated item rather than cash, so the payout structure stays intact.