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A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

• If the card is a face card, and the coin lands on heads, you win \(6

• If the card is a face card, and the coin lands on tails, you win \)2

• If the card is not a face card, you lose $2,no matter what the coin shows.

a. Find the expected value for this game (expected net gain or loss).

b. Explain what your calculations indicate about your long-term average profits and losses on this game.

c. Should you play this game to win money?

Short Answer

Expert verified

(a) The expected loss is -813.

(b) Calculations indicate that in the long term there will be a loss.

(c) No, I should not play this game.

Step by step solution

01

Given information (part a)

If a card is a face card and the coin lands on the head, then you win $6.

If a card is a face card and the coin lands on tails, then you win $2.

If a card is not a face card, then you lose $2.

02

Explanation (part a)

X
Px
XPx
6
1252×12=326
6×326=913
2
1252×12=326
2×326=313
-2
4052×1=1013
-2×2013=-313
Total1
-813

Ex=X×PxEx=6×913+2×313-2×-2013Ex=-813

There is an Expected loss of-8/13.

03

Given information (part b)

If a card is a face card and the coin lands on the head, then you win $6.

If a card is a face card and the coin lands on tails, then you win $2.

If a card is not a face card, then you lose$2.

04

Explanation (part b)

This indicates that in long term there will be a loss because the probability of getting a loss is more than getting a profit. So, there will be a loss.

05

Given information (part c)

If a card is a face card and the coin lands on the head, then you win$6.

If a card is a face card and the coin lands on tails, then you win $2.

If a card is not a face card, then you lose$2.

06

Explanation (part c)

NO, I should not play this game as we have already seen in the above part that there is a higher probability of loss. So, we will not play this game.

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