Suppose a game starts with

$3$

$$pound coins in the pot. At each turn the

number of coins in the pot is counted and the following procedure is repeated K times:

a die is thrown, and according to the outcome the following four things can happen

$$

$$If the outcome is

$1$

$$or

$2$

$$the player takes

$1$

$$coin from the pot.

$$

$$If the outcome is

$3$

$$nothing happens.

$$

$$If the outcome is

$4$

$$the player puts

$1$

$$extra coin in the pot

$($

you may assume that

the player has unlimited supply of coins

$)$

$.$

$$

$$If the outcome is

$5$

$$or

$6$

$,$

$$the player puts

$2$

$$extra coins in the pot.

If there are no coins remains in the pot, the game stops.

$($

a

$)$

$$Compute the expected number of coins in the pot after turn n in N

$.$

$[$

$4$

$]$

$($

b

$)$

$$Compute the probability that the game will stop eventually.

$[$

$4$

$]$

$($

c

$)$

$$Let mn be the maximal possible number of coins in the pot after the n

$-$

th turn?

$[$

$7$

$]$

What is the probability that the actual number of coins in the pot after n turns

is equal to mn

$$

$1$

$?$