Harold and Timmy are playing a game of chance.In each round of the game, both players roll a fair die.If the sum of the numbers on both dice is less than or equal to 6, Harold is considered the "winner." If the sum of the numbers on both dice is greater than or equal to 7, Timmy is considered the "winner."

Whoever loses picks a card from a standard deck where all Aces have been removed and must pay the winner according to the table below.After the payment has been made, they roll the dice again to start the next round.

Each player starts the game with $10.Assume all die rolls and card draws are independent.

+-------------+-------+---------+-----

| Card Drawn|2-10 | J-K |

+=============+======

| Amount | $1 | $5 |

+-------------+----------+---------+--

a) Is this a fair game? Explain.

b) What distribution best models the number of times Harold wins in 20 rounds of the game?Be specific.If it is a parametric distribution, state the parameters.

c) Let X = amount won in a given round.What is the distribution of X (i.e. define P(X = x) for all possible outcomes)?

d) If Timmy wins in round 1, what are his expected earnings?

e) Let X = Amount of money earned by Harold in one round of the game (when Harold loses to Timmy his earnings are negative).

i. What is the sample space of X?

ii. Give the probability mass function of X.

iii. Calculate E(X)

iv. Calculate Var(X)

f) What is the probability that one of them is broke after 2 rounds of the game?