Charity Fair: Is the Token Game Worth Running?
A charity fair is planning a token game. A bag contains 40 tokens: 4 red, 10 blue and 26 white. A player pays €2 to draw one token. A red token wins €5, a blue token wins €1, and a white token wins nothing. After each draw the token is put back, so the probabilities stay the same. The organisers expect 200 plays during the day. They need the game to raise at least €300 after prizes, because the hall hire is €300. They can change either the entry price or one prize, but not both, and they want to keep the game simple. Before deciding, they ask for the probability of each result, the expected profit per play and over 200 plays, and one realistic change that would meet the €300 target. All prizes are paid in cash from the takings.
1. Find the probability of drawing red, blue and white. Check that the probabilities sum to 1.
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There are 40 tokens in total.
P(red) = 4/40 = 0.1
P(blue) = 10/40 = 0.25
P(white) = 26/40 = 0.65
Check: 0.1 + 0.25 + 0.65 = 1.0, so the probabilities are valid and cover every possible result.
2. Calculate the expected prize payout per play and the expected profit over 200 plays at the current €2 price.
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Expected payout per play = (probability of red × €5) + (probability of blue × €1) + (probability of white × €0).
= (0.1 × 5) + (0.25 × 1) + (0.65 × 0)
= 0.50 + 0.25 + 0
= €0.75 per play.
Expected profit per play = entry price − expected payout = 2.00 − 0.75 = €1.25.
Over 200 plays: 200 × 1.25 = €250.
The game is expected to raise €250, which is €50 short of the €300 target.
3. Suggest one change to the entry price or one prize that would raise the expected profit to at least €300 over 200 plays. Show the new expected profit.
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One simple change is to lower the red prize from €5 to €2.
New expected payout = (0.1 × 2) + (0.25 × 1) + (0.65 × 0)
= 0.20 + 0.25 = €0.45 per play.
New profit per play = 2.00 − 0.45 = €1.55.
Over 200 plays: 200 × 1.55 = €310.
This meets the €300 target with €10 to spare.
Another valid change is to raise the entry price to €2.25 while keeping prizes the same. Then profit per play = 2.25 − 0.75 = €1.50, and over 200 plays = €300 exactly. Either change is simple and reaches the target.