Three black balls and two white balls are placed in a bag. All the balls are indistinguishable apart from their colors.
Mr Black and Mr White take turns to draw a ball from the bag at random with replacement. The first player to draw the ball whose color matches his name wins the game and the game stops immediately. The games goes on until a winner is found.
If Mr.Black draws first,
a)find the probability that Mr Black wins the game at his third draw.
b)find the probability that Mr White wins the game at his third draw.
c)find the probability that the winner is found at his n draw. (Express your answer in terms of n).
d)Show that the probability that Mr White wins the game is 4/19. (Hint: If 0<p<1, then 1+p+p^2+p^3 +… =1/1-p)
After playing the game for a few rounds. Mr White realizes that he loses most of the time, as he has fewer balls to begin with and Mr Black always draws first. He then proposes to change the rule of the game slightly: Use a fair, six-sided die to decide who draws first. If the outcome is “6”, Mr Black draws first. Otherwise, Mr White draws first.
e)With this new rule, find the probability that Mr White wins the game.