Answer:
$1.68 per match
Average Fee = $1.69
Step-by-step 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.