Carnival Game: Weighted Probability and Expected Value
At a school carnival, a club plans a game with a fair six-sided die and two bags of marbles. If the die shows 1, 2, 3, or 4, the player draws one marble from Bag A, which contains 2 red and 3 blue marbles. If the die shows 5 or 6, the player draws one marble from Bag B, which contains 4 red and 1 blue marble. A player wins by drawing blue. The club plans to charge $2 per play and award a $3 prize for a win. A student volunteer says the win probability is 1/2 because one bag has more blue than red and the other has more red than blue. The club advisor wants a correct probability, the expected net gain or loss per play for the player, and a fair price for the game if the club wants the expected payout to equal the price.
1. Compute the probability that a player wins. Use conditional probabilities or a tree diagram and explain each branch.
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The die determines which bag is used, so the branches are weighted by the die probabilities.
Branch 1: The die shows 1, 2, 3, or 4. That has probability 4/6 = 2/3. Given that, Bag A has 3 blue marbles out of 5 total, so P(blue | Bag A) = 3/5. This branch contributes (2/3)(3/5) = 6/15 = 2/5.
Branch 2: The die shows 5 or 6. That has probability 2/6 = 1/3. Given that, Bag B has 1 blue marble out of 5 total, so P(blue | Bag B) = 1/5. This branch contributes (1/3)(1/5) = 1/15.
Total win probability = 2/5 + 1/15 = 6/15 + 1/15 = 7/15, which is about 0.4667, or 46.67%. The student's 1/2 is not correct because the two bags are not equally likely and the conditional probabilities must be weighted by the die outcomes.
2. Calculate the expected prize payout per play and the player's expected net gain if the charge is $2. Should the player expect to win or lose money per play in the long run?
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The expected prize payout is the prize amount times the win probability: (7/15) × $3 = 21/15 = $1.40.
The player pays $2.00 per play, so the player's expected net gain is expected payout minus cost: $1.40 − $2.00 = −$0.60.
In the long run, the player should expect to lose about $0.60 per play on average, assuming the game is played many times and the die and bags are fair. Equivalently, the club expects to collect about $0.60 per play before expenses.
3. What price would make the game fair in expected value? If the club wants to raise about $0.50 per play on average while keeping the $3 prize, what price should it charge, and what assumptions are you making?
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A game is fair in expected value when the player's expected net gain is zero, so the price should equal the expected payout. The expected payout is $1.40, so a price of $1.40 per play would make the game fair, ignoring other costs.
If the club wants to raise about $0.50 per play on average, set price − expected payout = $0.50. Thus price = $1.40 + $0.50 = $1.90. At $1.90 per play, the club's expected margin is $0.50 per play.
This assumes the $3 prize is the only payout, the die and marbles are fair, plays are independent, and operating costs are ignored. If the club charges $2.00 instead, its expected margin is $2.00 − $1.40 = $0.60 per play.