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A local sports club is planning to run a weekly jackpot - Leaving Cert Mathematics - Question 6 - 2016
Question 6
A local sports club is planning to run a weekly jackpot. To win the Jackpot of €1000, contestants must match one letter chosen from the 26 letters in the alphabet an... show full transcript
Worked Solution & Example Answer:A local sports club is planning to run a weekly jackpot - 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 calculate this probability, we consider that there are 26 letters and the numbers range from 0 to 9. The selection of the letter M and the numbers 3 and 3 can be expressed as follows:
The total possible outcomes for the letter and numbers are represented as:
Thus, the probability that M, 3, 3 is the winning outcome is ( \frac{1}{2600} ).
Step 2
If a contestant matches the letter only, or the letter and one number (but not both numbers), they will win €50. Using the table below, or otherwise, find how much the club should expect to make or lose on each play.
Answer
To calculate the expected loss or gain per play:
-
Win Jackpot:
- Payout: €1000
- Probability: ( P(Matching) = \frac{1}{2600} )
- Expected Value: ( 1000 \times \frac{1}{2600} = \frac{1000}{2600} \approx 0.385 )
-
Match letter and first number only:
- Payout: €50
- Probability: ( P = \frac{9}{2600} )
- Expected Value: ( 50 \times \frac{9}{2600} = \frac{450}{2600} \approx 0.173 )
-
Match letter and second number only:
- Same calculation as above, so ( 0.173 ).
-
Match letter and neither number:
- Payout: €50
- Probability: ( P = \frac{81}{2600} )
- Expected Value: ( 50 \times \frac{81}{2600} = \frac{4050}{2600} \approx 1.558 )
-
Fail to win:
- No payout, probability: ( P = 1 - (sum \ of \ previous \ probabilities) = \frac{2500}{2600} )
- Expected Value: ( 0 )
Total expectation:
Sum up all expected values:
Since the club charges €2, the club loses approximately 29 cents per play.
Step 3
If the club wants to make an average profit of €600 per week from the lotto, how much should the club charge per play?
Answer
To find the price per play, we set up the equation:
Let the charge per play be ( x ).
The total income per week is ( 845 \times x ). The total payout for losses is formulated as:
In our case, for an average profit of €600:
Solving for ( x ):
- Calculate total payout:
( 2.288 \times 845 ) evaluating gives us around €1939.76. - Insert back into the equation:
( 845x - 1939.76 = 600 )
( 845x = 600 + 1939.76 )
( 845x = 2539.76 ) ( x = \frac{2539.76}{845} \approx 3.00 )
Thus, the club should charge approximately €3.00 per play.