P1.3 An urn contains 3 red, 2 white, and 1 black balls. Law and Robin take turns

Question

P1.3 An urn contains 3 red, 2 white, and 1 black balls. Law and Robin take turns and draw balls from this urn one at a time, with no repetition (i.e. drawn balls are not put back in the urn). First person to draw a white ball wins. And if a person draws the black ball, that person loses automatically. Assuming that all the balls in the urn are equally likely to be drawn at any stage in this game and Law is the first person to draw a ball; (a) determine the probability of Law winning the game. (b) Determine the probability of Law winning the game, given that the first ball he drew was red. (c) Determine the probability of game ending with someone drawing the black ball. (d) Determine the probability of Robin drawing the black ball, given that Law won the game.

Explanation / Answer

(a) Here law will win the game if he draw the white ball first and not drawing the black one.

so here if we count Red as R , white as W and black as B then he will win when he get

1W (that means first chance white) or 1R 2B (means he draw red and robin draw black), 1R2R3W, 1R2R3R4B

so probability of that happening = Pr(First draw is white) + Pr(First draw is Red and robin draw black) + Pr(first two draw are red and third draw is White) + Pr(First three draw is Red and fourth draw is black)

= 2/6 + 3/6 * 1/5 + 3/6* 2/5* 2/4 + 3/6 * 2/5 * 1/4 * 1/3

= 0.55

(b) Here given that the first ball is red drawn then possibilities are

(i) Robin draw a black in second chance (Robin lost) = 1/5

(ii) Robin draw a white in second chance (Robin Won) = 2/5

(iii) Robin draw a Red in second chance (game proceeds) = 2/5

(a) Law draw a White (Law wins) = 2/4

(b) Law draw a Black(Robin wins) = 1/4

(c) Law draw a Red(GameProceeds) = 1/4

(1) Robin Draw a black (Game ends Law wins) = 1/3

(2) Robin draw a white (Game ends robin wins) = 2/3

Pr(Law wins) = 1/5 + 2/5 * 2/4 + 2/5 * 1/4 * 2/3 = 0.4667

(c) Here now we have to find the probability that game ends with someone drawing the black ball. Possibilities are

(i) Law first chance black ball = 1/6

(ii) Law first draw red robin second draw black ball = 3/6 * 1/5

(iii) Law first draw red robin second draw red Law third draw black ball = 3/6 * 2/5 * 1/4

(iv) initially three draw red and Robin fourth draw black = 3/6 * 2/5 * 1/4 * 1/3

so

game ends with black ball = 1/6 + 3/6 * 1/5 + 3/6 * 2/5 * 1/4 + 3/6 * 2/5 * 1/4 * 1/3 = 1/3

d) Here law won the game so robin draw the black ball in second or in fourth draw

so

Pr(robin drawing black ball l law won) = (3/6 * 1/5 + 3/6 * 2/5 * 1/4 * 1/3)/ 0.55 = 0.2121

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