Question

Your friend approaches with three, fair 5-sided dice (labeled 1 to 5). You get to roll them and your friend gives you money based on your results. If all three faces match (e.g., 2.2.2), you win $6. If only two faces match (e.g., 5, 1, 5), you win $2. If no faces match (e.g., 3, 4, 2), you win just $1.

a. Find the expected payout from your friend. WISE
b. Your friend is losing a lot of money and decides to charge you a fee each time you play. What fee should the friend change if she wants to earn 10 cents, on average, each time you play?
c. (Ignore the fee for playing in this part.) Instead of playing once a day for a week (seven days), your friend suggests you just play once a week and that all payouts are simply multiplied by 7. Give two reasons why your friend's plan is not the same as playing once a day for seven days.

Answer 1

$1.68 per match

Average Fee = $1.69

Explanation:

Given:

- Earnings for all 3 different number = $ 1

- Earnings for 2 same number = $ 2

- Earnings for 3 same number = $ 6

- Five sided fair dice = 3

Find:

- Expected pay-out

- What fee charged per match for 10 cent income

- Reason for change to new match

Solution:

- Construct a probability distribution table, where X: payout per match

X $1 $2 $6

P(X) 0.48 0.48 0.04

Case X = $1 : _ _ _ All three different numbers

No.outcomes 5 * 4 * 3 = 60

Total outcome 5 * 5 * 5 = 125

Hence, P(X= $ 1) = 60 / 125 = 0.48

Case X = $2 : S _ S two numbers are same

No.outcomes = 5 * 4 * 1 = 20 per combination

Total combinations = S _ S + S S _ + _ S S = 3

Total outcome = 3 * 20 = 60

Hence, P(X= $ 2) = 60 / 125 = 0.48

Case X = $6 : S S S two numbers are same

No.outcomes = 5 * 1 * 1 = 5 per combination

Total combinations = 1

Total outcome = 5 * 1 = 5

Hence, P(X= $ 2) = 5 / 125 = 0.04

- Expected Payout E(X):

E(X) = 1*0.48 + 2*0.48 + 6*0.04 = $1.68 per match

- To earn $0.01 on average the fee of match is:

Fee_avg = E(X) + $0.01 = $1.69

- With this new plan your friend would loose less. The expected payout of a match is $1.68 for which the probability X < $ 2 is around = 0.48. However, to gain a $2 or higher P ( X > 2 ) = 0.48 + 0.04 = 0.52. Hence, the week-payout for X > 2 is greater than 7*2 = $14. So probability of week's payout to be $14 is higher than to be $ 11.76 as per average pay per day.