Question

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 $9
•	If the card is a face card, and the coin lands on Tails, you win $3
•	If the card is not a face card, you lose $3, no matter what the coin shows.

• Part (a)

  Find the expected value for this game (expected net gain or loss). (Round your answer to two decimal places.)
  $

• Part (b)

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

• The calculated value represents a fixed amount that your total money will change after each loss.

• The calculated value represents the average amount per game that your total money will change over a large number of games.    

• The calculated value represents the average amount per loss that your total money will change over a large number of games. The calculated value represents a fixed amount that your total money will change after each game.

• Part (c)

• Should you play this game to win money?

• Yes, because the expected value indicates an expected average gain.

• No, because the expected value indicates an expected average loss.

Answer 1

Solution :-

Given data:

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 $9
• If the card is a face card, and the coin lands on Tails, you win $3
• If the card is not a face card, you lose $3, no matter what the coin shows.

(a) :-

Here we have to find out the expected value for this game.

We know that, In standard deck of cards, there exists total of 52 cards and in which 12 of face cards.

Number of no face cards = 52-12 = 40

Event of card	Net gain or loss	P(X)
Face card and head	$9	= (1/2) * (12 / 52) = 3 / 26
Face card and tail	$3	= (1/2) * (12 / 52) = 3 / 26
No face card and H or T	-$3	= (1) * (40 / 52) = 20 / 26

Then, Expected value is,

Expected value = (9) (3 / 26) + (3) (3 / 26) + (–3) (20 / 26)

Expected value =

Expected value =  

Expected value =  

Expected value = - $ 0.923

Expected value for this game = = -$ 0.923

(b) :-

Here we can say that, If you play this game repeatedly, over a long string of games, you would expect to lose $0.923 per game, on average.

Correct answer is, "The calculated value represents the average amount per loss that your total money will change over a large number of games".

(c) :-

Correct answer is, " No, because the expected value indicates an expected average loss".