Financial Expectation

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#7545

Expected Value & Frequency

Expected Value= The sum of the (probability × outcome)

Expected frequency = probability of event occurring × number of trials

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#14826

It costs $12 to play a game. There is a probability of $$\frac14$$ of winning $40 and a probability of $$\frac38$$ of winning $25. Calculate the financial expectation of this game.

P(lose) = $$1-\frac14-\frac38=\frac38$$

E = $$\frac14\times\,$40+\frac38\times\,$25-\frac38\times\,$10-$12$$    (don’t forget to subtract the cost of playing the game)

= $3.625

∴ E = $3.63

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#14827

A game is played by tossing two coins. The rules are: If two heads are thrown you win $10. If two heads are not thrown you lose $5.
a. Determine the financial expectation for this game.
b. If you played this game 50 times, how much would you expect to win or lose?

a. P(2 heads) = ¼  P(not 2 heads) = ¾

E = ¼ × 10 + ¾ × -5
E = -1.25
Lose $1.25

b. n = 50
E = 50 × -1.25

= -62.5

∴ lose $62.50

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#14828

In a raffle, 1000 tickets are sold costing $2 each. First prize is $350, second prize is $175, third prize is $100 and fourth prize is $50. No ticket can win more than one prize. What is the probability of winning first prize? What is the expected financial return?

P(win first prize) = ¹/₁₀₀₀
P(1st) = ¹/₁₀₀₀
P(2nd) = ¹/₁₀₀₀
P(3rd) = ¹/₁₀₀₀
P(no prize) = ⁹⁹⁷/₁₀₀₀

E = ¹/₁₀₀₀ × 250 + ¹/₁₀₀₀ × 125 + ¹/₁₀₀₀ × 100 + ⁹⁹⁷/₁₀₀₀ × -2
E = -1.519
∴ lose $1.52

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#14829

Amy buys a $1 ticket in a raffle. There are 200 tickets in the raffle and two prizes. First prize is $100 and second prize is $50. Find Amy’s financial expectation.

1st prize = ¹/₂₀₀ × 100
= $0.50
2nd prize = ¹/₂₀₀ × 50
= $0.25
No prize = ¹⁹⁸/₁₀₀ × -1
= -$0.99

total = 50 + 25 – 99 = -24

∴ loss of $0.24

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#18893

There is a game where there is a 20% chance of winning $50, 50% chance of winning $5 or losing $5 if no prize is won. What is the financial expectation of this game?

P(lose) = 0.3

E = 0.20 × $50 + 0.50 × $5 + 0.30 × -$5

= $11

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#18895

A coin is biased to that there is a 60% chance it will be heads when it is tossed.  If the coin is tossed and the result is “a head” you win $12 if the result is “a tail” you lose $15. Calculate the expected financial return.
P(H) = 0.6  win $12 P(T) = 0.4  lose $15	To obtain the Expectation (E) you must multiply the probability of the outcome by the financial return, remembering if it is a loss, then the number is negative
E = 0.6 × 12 + 0.4 × -15 = $1.20	so the expectation of the game is positive, which is a win, of $1.20

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#18898

John plays a game in which he has a 20% chance of winning $50, a 50% chance of winning $10 and a 30% chance of losing $5. What is John’s financial expectation when playing this game?
P($50) = 0.2 P($10) = 0.5 P(-$5) = 0.3	each probability must be multiplied by the outcome, and the loss will be a negative outcome
E = 0.2 × 50 + 0.5 × 10 + 0.3 × -5 E = $13.50	calculate the expectation, which in this case is winning  $13.50

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