A lottery offers one $800 prize, one $400 prize, and ten $80 prizes. One thousand tickets are sold at $3 each. Find the expectation if a person buts two tickets. Assume that each ticket can win more than one prize.
A lottery offers one $800 prize, one $400 prize, and ten $80 prizes. One thousand tickets are sold at $3 each. Find the expectation if a person buts two tickets. Assume that each ticket can win more than one prize.
To calculate the expectation, we need to find the probability of winning each prize multiplied by the value of each prize, and then sum them all up.
Let's break it down step by step:
Step 1: Calculate the probability of winning each prize.
The total number of tickets sold is 1,000, and a person is buying 2 tickets. So, the probability of winning each prize can be found by dividing the number of tickets bought (2) by the total number of tickets sold (1,000):
Probability of winning the $800 prize = 2/1,000 = 1/500
Probability of winning the $400 prize = 2/1,000 = 1/500
Probability of winning each $80 prize = 2/1,000 = 1/500
Step 2: Calculate the value of each prize.
The value of each prize is given in the question. We have:
Value of the $800 prize = $800
Value of the $400 prize = $400
Value of each $80 prize = $80
Step 3: Calculate the contribution of each prize to the expectation.
To find the contribution of each prize to the expectation, we multiply the probability of winning each prize by the value of each prize:
Contribution of the $800 prize = (1/500) * $800
Contribution of the $400 prize = (1/500) * $400
Contribution of each $80 prize = (1/500) * $80
Step 4: Sum up the contributions to find the expectation.
Finally, we sum up all the contributions to calculate the expectation:
Expectation = Contribution of the $800 prize + Contribution of the $400 prize + (Contribution of each $80 prize * 10)
Expectation = ((1/500) * $800) + ((1/500) * $400) + ((1/500) * $80 * 10)
You can simplify this expression to find the final expectation.