The Question
A game at a fun park requires a player to roll a six-sided dice and pick a card from a deck of 52 cards. A player wins if an odd number appears on the uppermost face of the dice and the player also draws a picture card from the deck. A deck of cards has 4 suites (hearts, diamonds, spades and clubs). There are 4 picture cards (king, queen, jack and ace) in every suite. A player pays R10 to play a game and in an hour, 260 people can each play one game. If the owner wants to make a 70% profit per hour, calculate the maximum amount that the owner must pay out to each winner.
Details
💡 Hint
Break the problem into two main parts: first, calculate the probability of winning. Second, calculate the financial aspects – how much money is available for payouts after the owner takes their profit? Don’t forget to consider the number of players!
📝 Solution Steps
- 1. Calculate the probability of rolling an odd number on a six-sided die.
- 2. Calculate the probability of drawing a picture card from a standard deck of 52 cards.
- 3. Calculate the overall probability of winning (P(odd) * P(picture card)).
- 4. Calculate the total money collected from 260 players.
- 5. Calculate the owner’s desired profit (70% of total collected).
- 6. Determine the maximum total amount available for payouts (Total collected – Profit).
- 7. Calculate the expected number of winners (Total players * Probability of winning).
- 8. Calculate the maximum amount the owner must pay out to each winner (Total payout / Expected number of winners).
📚 Explanation
This problem combines probability with financial calculations. First, determine the probability of a player winning. This involves two independent events: rolling an odd number on a die and drawing a picture card. Multiply their individual probabilities to get the overall winning probability. Next, calculate the total money collected from all players. Then, determine the owner’s desired profit (70% of the total collected) and subtract it to find the maximum total amount available for payouts. Finally, estimate the number of winners (total players multiplied by the winning probability) and divide the total payout amount by this number to find the maximum payout per winner.
✅ Answer
R19.50
⚠️ Common Mistakes
- Incorrectly identifying the number of odd outcomes or picture cards.
- Confusing independent events (multiplication) with mutually exclusive events (addition).
- Errors in calculating the profit percentage or the total amount available for payouts.
- Not correctly determining the expected number of winners.
- Arithmetic errors in multi-step calculations.
📐 Formulas Required
- P(A and B) = P(A) * P(B) (for independent events)
- Profit = Revenue – Cost
- Percentage calculations