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A local sports club is planning to run a weekly lotto - Leaving Cert Mathematics - Question 6 - 2016
Question 6
A local sports club is planning to run a weekly lotto. To win the Jackpot of €1000, contestants must match one letter chosen from the 26 letters in the alphabet and ... show full transcript
Worked Solution & Example Answer:A local sports club is planning to run a weekly lotto - Leaving Cert Mathematics - Question 6 - 2016
Step 1
Calculate the probability that M, 3, 3 would be the winning outcome in a particular week.
Answer
To find the probability of getting the outcome M, 3, 3 in this lotto, we need to consider the total possible outcomes for letter and numbers. There are 26 letters and 10 possible numbers (0-9).
- Choose one letter: There are 26 options.
- Choose first number: There are 10 options.
- Choose second number: There are 10 options (numbers can be repeated).
Thus, the total number of combinations is:
The probability of winning M, 3, 3 specifically is:
P(M, 3, 3) = rac{1}{2600}
So, the final answer is:
P(M, 3, 3) = rac{1}{2600}
Step 2
Find how much the club should expect to make or lose on each play, correct to the nearest cent.
Answer
The expected values for each event must be calculated based on the payouts and their probabilities. The payout for winning the jackpot is €1000, while matching one letter and one number yields €50.
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Win Jackpot (1 outcome):
- Payout: €1000,
- Probability: rac{1}{2600}
- Expected Value: 1000 imes rac{1}{2600} = rac{1000}{2600} = 0.3846
-
Match letter and first number (1 outcome):
- Payout: €50,
- Probability: rac{1}{2600}
- Expected Value: 50 imes rac{1}{2600} = rac{50}{2600} = 0.0192
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Match letter and second number (1 outcome):
- Payout: €50,
- Probability: rac{1}{2600}
- Expected Value: 50 imes rac{1}{2600} = rac{50}{2600} = 0.0192
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Fail to win (the rest of the outcomes):
- Payout: €0,
- Probability: Remaining outcomes: 2597 outcomes,
- Expected Value: 0 imes rac{2597}{2600} = 0
Now sum these expected values:
Since they charge €2 per play, the expected profit or loss is:
Expected Profit/Loss = Charge - E = €2 - €0.423 = €1.577
Thus, on each play, the club expects to make approximately €1.58.
Step 3
how should the club charge per play, correct to the nearest cent?
Answer
To calculate how much the club should charge per play to achieve an average profit of €600 per week when the average number of plays is 845:
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Total Expected Profit Required per Week: €600
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Total Number of Plays per Week: 845
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Expected Profit per Play Required:
ext{Expected Profit per Play} = rac{600}{845} \\ ext{Expected Profit per Play} = 0.7088
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Charge per Play Calculation:
- If the club wants to make a profit of approximately €0.71 per play, then the total charge per play should be the sum of the expected profit per play and the base charge for each play of €2:
Rounding to the nearest cent, the club should charge:
€2.71 per play.