A lottery has a grand prize of $180,000, two runner-up prizes of $22,500 each, eleven third-place prizes of $3600 each, and fourteen consolation prizes of $450 each. If 720,000 tickets are sold for $1 each and the probability of any one ticket winning is the same as that of any other ticket winning, find the expected return on a $1 ticket. (Round your answer to two decimal places.)
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(1/720,000)*180,000 + (2/720,000)*22,500 +(11/720,000)*3600 + (14/720,000)*450 = $ 0.38
To find the expected return on a $1 ticket, we need to calculate the probability of winning each prize and multiply it by the value of that prize.
First, let's find the probability of winning the grand prize. Since there are 720,000 tickets sold and only 1 grand prize, the probability of winning the grand prize is 1/720,000.
Next, let's find the probability of winning a runner-up prize. There are 2 runner-up prizes, so the probability of winning a runner-up prize is 2/720,000.
Now let's find the probability of winning a third-place prize. There are 11 third-place prizes, so the probability of winning a third-place prize is 11/720,000.
Lastly, let's find the probability of winning a consolation prize. There are 14 consolation prizes, so the probability of winning a consolation prize is 14/720,000.
Now, let's calculate the expected return.
The expected return is calculated by multiplying the probability of winning each prize by the value of that prize and summing them up.
Expected return = (Probability of winning grand prize * Value of grand prize) + (Probability of winning runner-up prize * Value of runner-up prize) + (Probability of winning third-place prize * Value of third-place prize) + (Probability of winning consolation prize * Value of consolation prize)
Expected return = (1/720,000 * $180,000) + (2/720,000 * $22,500) + (11/720,000 * $3600) + (14/720,000 * $450)
Expected return = $0.25
Therefore, the expected return on a $1 ticket is $0.25.
To find the expected return on a $1 ticket, we need to calculate the probability of winning each prize and multiply it by the amount of that prize.
Step 1: Calculate the total amount of prize money:
- Grand prize: $180,000
- Runner-up prizes: $22,500 x 2 = $45,000
- Third-place prizes: $3,600 x 11 = $39,600
- Consolation prizes: $450 x 14 = $6,300
Total prize money: $180,000 + $45,000 + $39,600 + $6,300 = $270,900
Step 2: Calculate the probability of winning each prize:
The probability of winning the grand prize is 1 out of 720,000 since there is only one grand prize.
The probability of winning a runner-up prize is 2 out of 720,000 since there are two runner-up prizes.
The probability of winning a third-place prize is 11 out of 720,000 since there are eleven third-place prizes.
The probability of winning a consolation prize is 14 out of 720,000 since there are fourteen consolation prizes.
Step 3: Calculate the expected return on a $1 ticket:
- Expected return = (Probability of winning grand prize x Amount of grand prize) + (Probability of winning runner-up prize x Amount of runner-up prize) + (Probability of winning third-place prize x Amount of third-place prize) + (Probability of winning consolation prize x Amount of consolation prize)
- Expected return = (1/720,000 x $180,000) + (2/720,000 x $45,000) + (11/720,000 x $3,600) + (14/720,000 x $450)
- Expected return ≈ $0.25
Therefore, the expected return on a $1 ticket is approximately $0.25.