Probability with replacement

Question

A bag contains 8 black balls,3 red balls, and 9 white balls. if three balls are drawn at random with replacement. what's the probability that (a) the balls are all black (b) two balls are black and one is white (c). One ball is of each colour (d) None of the ball is red (e) either two balls is black and one is red (f) they are of the same colour

Answer 1

(a) There are 20 balls and 8 of them are black, so the probability that a randomly drawn ball is black is 8/20=2/5. If we replace the balls after drawing, the probability of each being black is independent of the preceding balls. Thus for (a) the probability is (2/5)*(2/5)*(2/5)=8/125.

(b) The probability that a ball is white is 9/20. There are three configurations in which we could draw two black balls and a white ball: WBB, BWB, BBW. Each of these has probability (2/5)*(2/5)*(9/20) = 9/125, and so the total probability of this event is 3*(9/125)=27/125.

(c) The probability of drawing a red ball is 3/20, so the probability of any individual drawing with three different colors is (2/5)*(9/20)*(3/20) = 27/1000. How many permutations of the letters B, W, R are there? There are 6: BWR, BRW, WBR, WRB, RBW, RWB.Thus the probability of this event is 6*(27/1000)=81/500.

(d) The probability that a given ball is not red is 17/20. By independence the probability that all three balls are not red is (17/20)^3 = 4913/8000.

(e) [I think there is a typo here and the correct problem statement is "either two balls are black OR one is red", where the "either" indicates exclusive or] The probability of two black balls is 3*(2/5)*(2/5)*(9/20) because there are three options for the non-black ball and the probability that it is not black is equal to the probability that it is white, which is 9/20 (here we are assuming that the "either" means that the third ball cannot be red. The probability that one ball is red (and two balls are not black) is 3*3/20*(9/20*9/20 + 2*9/20*3/20), where the 3 in front comes from the three choices as to which ball is red, the 9/20*9/20 corresponds to the case when the other two balls are white, and 2*9/20*3/20 corresponds to the other balls being black and white. This simplifies to 243/1600.

(f) They are of the same color if they are all red, all black, or all white. The probability of all red is (3/20)^3, the probability of all black is (2/5)^3, and the probability of all white is (9/20)^3, so the probability of this event is

(3/20)^3+(2/5)^3+(9/20)^3 = 317/2000.