Probability!!

Question

1. At a raffle, 3000 tickets were sold at $5 a piece for four different prices of $1500, $1000, $500, and $100. Build a probability distribution table with all assigned probabilities. Calculate the expected value of this game. Would you buy one of these tickets? Why or why not?

 

2. A gambler rolls 2 dice. He wins 15 dollars if the sum of the two dice adds up to “lucky seven”. The cost to play this game is $10. What is the expectation of this game? Will you play this game? Why or why not? Hint: you need to know how many outcomes from tossing 2 dice will add up to 7; check lecture slides 4.2.

 

3. In a “draw one” card game, if you draw one card and get a heart, you will make $10. If you draw a face card (with the exception of hearts) you will make $8. If you draw anything else, you will lose $15. What is the expectation of this game? Would you play this game? Why or why not? Hint: a regular card deck has 52 cards, 13 hearts, and 12 face cards.

Answer 1

Hi Jack,
1.
Probability Table for payoff from one ticket
Outcome    Probability Expected payoff (= outcome x probability of outcome)
0                    2996/3000                                             0
100                      1/3000                         100/3000 = 1/30
500                      1/3000                         500/3000 = 1/6
1000                    1/3000                       1000/3000 = 1/3
1500                    1/3000                       1500/3000 = ½

Expected payoff = 1/30 + 1/6 + 1/3 + ½ = 31/30 ≈ 1.03

No, I would not buy because the game has an expected payoff of about $1.03 for a $5 ticket.

2.
There are 36 possible outcomes, 6 possible outcomes on the first roll and for each of those 6 outcomes, 6 possible outcomes on the second roll. Each of these outcomes is equally likely, so we will enumerate the number of ways a total of 7 can be made for the 2 rolls:

First roll Second roll
1 6
6 1
2 5
5 2
3 4
4 3

From the six possible ways to make a 7 as the sum of two rolls, we know the probability of making a 7 as the sum of two rolls is 6/36 = 1/6. With a probability of 5/6, some other outcome that does not add up to 7 is obtained. The payoff is 0 for the outcomes not equal to 7 and 10 for the outcome that the rolls add to 7, so the expected outcomes is

(5/6 x 0) + (10 x 1/6) = 10/6 ≈$1.67

No, I would not play this game because it costs $10 and has an average payout of $1.67.

3.
We need to know the probability of drawing a heart and the probability of drawing a face card that is not a heart. Each draw is equally likely and there are 13 hearts and 52 cards in a deck, so the probability of a heart is 13/52 = ¼. There are 12 face cards in the deck but only 9 face cards that are not hearts and so 9/52 chance of getting a face card that is not a heart. The rest of the outcomes that pay -15 eat up the rest of the probability, 30/52. Therefore the expected payoff off the game is

(¼ x 10) + (9/52 x 8) + (30/52 x -15) ≈ -$4.77

No, I would not play this game because the expected payoff is a loss of about $4.77.

Hope this helps, feel free to ask questions.

Kind regards,
Hugh