Question

Someone sells 2000 tickets for $1 each. Prizes are awarded of one $100, four $50, and eight $25.
Find the expected value if you purchase 1 ticket. Your expected value should end up negative since there are way more chances to not win than to win one of the prizes.
Expected value is calculated by multiplying every possible outcome by its probability and then adding those products. Let's break this down.
a) In this case there are 2000 outcomes. How many expect to be out their dollar and not win anything?
b) What is the probability of not winning anything?
c) How many will win $25?
d) What is the probability of winning $25?
e) If you win $25 after spending $1 for the ticket, what is your true gain?
f) How many will win $50?
g) What is the probability of winning $50?
h) If you win $50 after spending $1 for the ticket, what is your true gain?
i) How many will win $100?
j) What is the probability of winning $100?
k) If you win $100 after spending $1 for the ticket, what is your true gain?
l) Finally add up all the probabilities times the gain or loss associated with them. What is the expected value if you purchase one ticket?
m) What can you expect to happen if you purchase 5 tickets?

Answer 1

(a) The total number of prizes = 1 + 4 + 8 = 13

Therefore 2000 - 13 = 1987 expect to not win anything

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(b) The probability of not winning anything = 1987 / 2000 = 0.9935

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(c) Number of people who will win $25 = 8

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(d) Probability of winning $25 = 8 / 2000 = 0.004

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(e) True Gain on winning $25 = 25 - 1 = $24

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(f) Number of people who will win $50 = 4

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(g) Probability of winning $50 = 4 / 2000 = 0.002

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(h) True Gain on winning $50 = 50 - 1 = $49

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(i) Number of people who will win $100 = 1

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(j) Probability of winning $100 = 1 / 2000 = 0.0005

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(k) True Gain on winning $100 = 100 - 1 = $99

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(l) Therefore Sum of probabilities = (24 * 0.004) + (49 * 0.002) + (99 * 0.0005) - (1 * 0.9935)

0.096 + 0.098 + 0.0495 - 0.9935 = -0.75 = $- 75 cents

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(m) If you purchase 5 tickets, the expected gain = -0.75 * 5 = $-3.75

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There is a shorter method, where we can do away with all these calculations.

The total amount collected = 2000 * 1 = $2000

The total Amount given Away in prizes = (100 * 1) + (50 * 4) + (25 * 8) = 100 + 200 + 200 = $500

Net Profit (Overall) = 2000 - 500 = $1500
Net Profit / Ticket = 1500 / 2000 = 0.75

The net profit / ticket to them is my net loss or expected value = -0.75

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