Manuel has a deck of 10 cards numbered 1 through 10. He is playing a game of chance.
This game is this: Manuel chooses one card from the deck at random. He wins an amount of money equal to the value of the card if an odd numbered card is drawn. He loses $4.20 if an even numbered card is drawn.
What can Manuel expect in the long run, after playing the game many times?
(He replaces the card in the deck each time.)
Manuel can expect to gain money.
He can expect to gain __ dollars per draw
Manuel can expect to lose money.
He can expect to lose __ dollars per draw
Manuel can expect to break even (neither gain nor lose money).
prob(to pick a specific card) = 1/10
expectation of win
= (1/10)(1+3+5+7+9) = 25/10 = 2.50
expectation of loss
= (1/10)(2+4+6+8+10) = 30/10 = 3.00
bad game to play!
Manuel can expect to lose money.
He can expect to lose $2.90 per draw.
Why? Well, since there are an equal number of odd and even cards (5 each), Manuel has a 50% chance of drawing an odd card and winning the corresponding amount of money. On the other hand, he also has a 50% chance of drawing an even card and losing $4.20.
The expected value can be calculated as follows:
(0.50 * amount won) - (0.50 * amount lost)
So, (0.50 * $5) - (0.50 * $4.20) = $2.50 - $2.10 = $0.40
So, Manuel can expect to lose $0.40 per draw.
But since Manuel has to pay $4.20 each time an even card is drawn, we subtract that amount from his expected value:
$0.40 - $4.20 = -$3.80
So, Manuel can expect to lose $3.80 per draw.
But wait, there's more! Since we assumed he replaces the card in the deck each time, Manuel will have to pay $4.20 for every even card drawn, regardless if he has already paid for that card in a previous draw.
Taking that into account, Manuel can expect to lose $2.90 per draw in the long run.
Ouch! It's a humorously bad luck game for him.
To determine what Manuel can expect in the long run, we can calculate the expected value of each outcome.
There are 5 odd-numbered cards (1, 3, 5, 7, 9) and 5 even-numbered cards (2, 4, 6, 8, 10) in the deck.
The probability of drawing an odd-numbered card is 5/10 = 1/2, and the probability of drawing an even-numbered card is also 5/10 = 1/2.
When an odd-numbered card is drawn, Manuel wins an amount equal to the value of the card. The expected value of winning is therefore (1+3+5+7+9)/5 = 5.
When an even-numbered card is drawn, Manuel loses $4.20. The expected value of losing is -$4.20.
Now, we can calculate the expected value in the long run:
Expected value = (Probability of winning * Expected value of winning) + (Probability of losing * Expected value of losing)
Expected value = (1/2 * 5) + (1/2 * -4.20)
Expected value = 2.50 + (-2.10)
Expected value = $0.40
Therefore, Manuel can expect to break even in the long run (neither gain nor lose money), with an expected value of $0.40 per draw.
To determine Manuel's expected outcome in the long run, we need to compute the expected value of each possible scenario.
The probability of drawing an odd-numbered card is 5 out of 10 since there are 5 odd-numbered cards (1, 3, 5, 7, 9) out of a total of 10 cards.
The probability of drawing an even-numbered card is also 5 out of 10.
To calculate the expected value for the scenario where an odd-numbered card is drawn:
Expected Value = Probability of Winning x Amount of Money per Win
Expected Value = (5/10) x (Value of the Card)
For each odd-numbered card drawn, Manuel wins an amount of money equal to the value of the card. Therefore, the Expected Value for the scenario where an odd-numbered card is drawn is the average value of the odd numbers from 1 to 10:
Expected Value = (1 + 3 + 5 + 7 + 9) / 5 = 5.
To calculate the expected value for the scenario where an even-numbered card is drawn (causing Manuel to lose $4.20):
Expected Value = Probability of Losing x Amount of Money per Loss
Expected Value = (5/10) x (-$4.20) = -$2.10.
Now, to determine Manuel's expected outcome in the long run, we subtract the expected losses from the expected winnings:
Expected Outcome = Expected Value if Odd Card - Expected Value if Even Card
Expected Outcome = $5 - $2.10 = $2.90.
Therefore, Manuel can expect to gain $2.90 per draw in the long run.