Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing three green six yellow and five red marbles that both marbles are different colors.

Answer 1

3G, 6Y, 5R

Pr(2 of different color) = ?

first draw has 14 possible marbles, Pr(G)= 3/14, Pr(Y)=6/14 = 3/7, Pr(R)= 5/14

if it's Green, then odds the 2nd draw will not be green is 1-2/13 = 11/13

if it's Yellow, then odds the 2nd draw will not be yellow is 1-5/13= 8/13

if it's Red, odds the 2nd draw will not be red is 1-4/13 = 9/13

multiply and add them ups Pr(2 not the same color) = (3/14)(11/13) + (3/7)(8/13) + (5/14)(9/13) = 33/(13)(14) + 48/(13)(14) + (45/(14)(13) = 126/182

or alternatively, calculate the odds of getting 2 of the same color and subtract from 1:

Pr(G) on 1st draw is 3/14, 2nd draw 2/13, Pr(2G) = 6/182

Pr(Y) 1st time is 6/14, 2nd draw 5/13 Pr(2Y) = 30/182

Pr(R) 1st time is 5/14, 2nd 4/13 Pr(2R) = 20/182

Add them Pr(2 same color) = 56/182

subtract from 1: 1-56/182 = (182-56)/182 = 126/182

Pr(2 different colors) = 126/182 = 63/91 = about 69.23%