Question

Suppose you have a bag containing 2 red marbles, 3 white marbles, and 4 blue marbles.

a) if you draw 3 marbles simultaneously, what is the probability that you get one of each color?

b) If you draw 3 marbles one at a time, what Is the probability that you get one of each color?

Answer 1

Please note ⁿC_x = n! / [(n-x)!*x!]

Probability = Favourable Outcomes/Total Outcomes

Total Number of Marbles = 2R + 3W + 4B = 9

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(a) When drawing 3 marbles simultaneously, P(Getting one of each colour)

Favourable Outcomes = choosing one from 2 red, 1 froom 3 white and 1 from 4 Blue

= ²C₁ * ³C₁ * ⁴C₁ = 2 * 3 * 4 = 24

Total Outcomes = Choosing 3 out of 9 balls = ⁹C₃ = 84

The Required probability = 24/84 = 2/7 = 0.2857

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(b) When drawing 1 marbles at a time, we can do this either by (i) With Replacement or (ii) Without replacement. This will be clearly specified. I have done both below for your reference.

(i) With Replacement. We will assume that we are picking Red first, 2nd White and third Blue. The probability that we get will be multiplied into 3! ways = 6 ways as we can get the marbles in 6 different orders(RWB, RBW, WBR, WRB, BRW, BWR).

P(first ball is red) = favourable outcomes/Total outcomes = 2/9

P(second ball is white). The total outcomes remain the same, as the first ball has been replaced = 3/9 = 1/3

P(Third ball is blue). The total outcomes remain the same = 4/9

Therefore the probability for this order (RWB) = (2/9) * (1/3) * (4/9) = 8/243

Therefore the overall probability = (8/243) * 6 = 16/81 = 0.1975

(ii) Without Replacement. We will assume that we are picking Red first, 2nd White and third Blue. The probability that we get will be multiplied into 3! ways = 6 ways as we can get the marbles in 6 different orders(RWB, RBW, WBR, WRB, BRW, BWR).

P(first ball is red) = favourable outcomes/Total outcomes = 2/9

P(second ball is white). The total outcomes reduce by 1, as the first ball has not been replaced = 3/8

P(Third ball is blue). The total outcomes again reduce by 1 = 4/7

Therefore the probability for this order (RWB) = (2/9) * (3/8) * (4/7) = 24/504

Therefore the overall probability = (24/504) * 6 = 24/84 = 2/7 = 0.2857

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