A deck of cards has the aces and face cards removed, so that the numbered cards 2-10 remain. The player draws a card and is paid the value of the card. Each play costs $5. If you play the game 20 times, how much money would you expect to win or lose overall?
I started to make a probability table with the x and p(x) labeled but i don't know how to continue from here.
There are 9x4 cards in the deck. All are equally probable.
if you get a 2, you lose 3 dollars
3...lose 2 dollars
4..lose 1 dollar
5 break even
6 win 1
7 win 2
8 win 3
9 win 4
10 win 5
The mean win is 1.50 per play, in 20 plays, that is 30 dollars.
The prob of drawing any specific number is 4/36 = 1/9
expected return on one game
= expectation(2) + expectation(3) + ... + expectation(10)
= (1/9)(2) + (1/9)(3) + ... + (1/9)(10)
= (1/9)(2+3+4+...+10)
= (1/9)(45) = 5
so it will cost 20($5) or $100 to play the 20 games
the expected return is 20($5).
So you will break even.
2+3+4+..+10 = 54 , not 45
so the expected return is (1/9)(54) = $6
so cost to play 20 games = 100
return = 20(6) = 120
so you would expect a profit of $20 to play 20 games
To calculate the expected value, we need to determine the probability of each outcome (x) happening, and then multiply that probability by the associated value of that outcome. Let's break down the problem step-by-step.
Step 1: Determine the probability of each outcome.
Since there are only numbered cards 2-10 remaining in the deck, there are a total of 9 possible outcomes.
To calculate the probability of drawing each number, we need to consider that there are 9 cards remaining. The probability of drawing a specific number is therefore 1/9.
Step 2: Assign values to each outcome.
Since you are paid the value of the card you draw, the value associated with each outcome is simply the number on the card itself.
So the values assigned to each outcome are as follows:
Outcome 2: 2
Outcome 3: 3
Outcome 4: 4
Outcome 5: 5
Outcome 6: 6
Outcome 7: 7
Outcome 8: 8
Outcome 9: 9
Outcome 10: 10
Step 3: Calculate the expected value.
To calculate the expected value, multiply each outcome value by its corresponding probability, and sum up all the results.
Expected Value = (2 x P(2)) + (3 x P(3)) + (4 x P(4)) + (5 x P(5)) + (6 x P(6)) + (7 x P(7)) + (8 x P(8)) + (9 x P(9)) + (10 x P(10))
where P(x) denotes the probability of outcome x.
By plugging in the probability values we calculated in Step 1, we can calculate the expected value.
Given that each play costs $5 and you play the game 20 times, we can now determine how much money you would expect to win or lose overall.
Overall Expected Winnings = Expected Value * Number of Plays - Total Cost
Overall Expected Winnings = Expected Value * 20 - (5 * 20)
Simplifying further will give you the final answer.