Question 838795: A normal deck of cards has 52 cards. 12 of these are Face cards. 4 are Aces. Here is a game. You pick a card. It is an Ace- you win $10. If it is a face card you win $5. Otherwise you loose $3.
a) make up a probability distribution
b)what is the expected value of the game
c)if you play ten times what is the expectation
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! there are 52 cards in the deck.
there are 4 aces.
there are 12 face cards.
there are 36 cards that are anything else but aces and face cards.
probability that the card is an ace is 4/52.
probability that the card is a face card is 12/52
probability that the card is not an ace and not a face card is 36/52.
if you get an ace, you will win $10.00.
if you get a face card, you will win $5.00.
if you get anything else, you will lose $3.00.
since you win $10.00 each time you get an ace, and you get an ace 4/52 of the the time, then your expected value is equal to 4/52 * $10.00 which is equal to 77 cents. you are expected to win 77 cents each time you play.
since you win $5.00 each time you get a face card, and you get a face card 12/52 of the time, then your expected value is equal to 12/52 * $5.00 which is equal to 1 dollar and 15 cents. you are expected to win 1 dollar and 15 cents each time you play.
since you lose $3.00 each time you get anything else, and you get anything else 36/52 of the time, then your expected value is equal to 36/52 * $3.00 which is equal to 2 dollars and 8 cents. you are expected to lose 2 dollars and 8 cents each time you play.
your total expected value is the sum of what you are expected to win minus the sum of what you are expected to lose.
if the balance is positive, you win whatever is indicated.
if the balance is negative, you lose whatever is indicated.
your balance comes out to be 77 cents plus 1 dollar and 15 cents minus 2 dollars and 7 cents for a total expected value of minus 15 cents which means you are expected to lose 15 cents each time you play.
$10.00 is won 7.7% of the time.
$5.00 is won 23.1% of the time.
$3.00 is lost 69.2% of the time.
If you played 10 times, this is what would happen.
you would win 10 dollars .077 * 10 = .77 times for a total of .77 * 10 = 7.7 dollars.
you would win 5 dollars .231 * 10 = 2.31 times for a total of 2.31 * 5 = 11.55 dollars.
you would lose 3 dollars .692 * 10 = 6.92 times for a total of 6.92 * 3 = 20.76 dollars.
take your total winnings of 7.7 + 11.55 = 19.25. subtract your total losings of 20.76 and you have a net loss of 1.51 dollars.
divide that by 10 times that you played and your average loss each time you play is .151 dollars which is equal to 15.1 cents which rounds out to 15 cents.
the numbers check out so they're good.
your expected value is how much you can win times the probability of winning minus how much you can lose times the probability of losing.
the total probability must be equal to 1 or 100%.
that checks out here as well since 7.7% + 23.1% plus 69.2% gets you a total of 100%.
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