A jar contains 600 jelly beans, 100 red, 200 green, and 300 blue. These are drawn randomly from the jar, one at a time, without replacement. What is the probability that the first color to be exhausted is red?
Proof. There will be a final string of beans of the same color (the color of the last bean drawn), and then a bean preceding it of a different color. The desired probability p is that these two colors are blue and green or green and blue, respectively. The probability that the last bean drawn is blue is 300/600 = 1/2, and, no matter how long the final string of blue beans is, the probability that the bean of preceding color is green is 200/(100+200) = 2/3. For green and blue these numbers become 200/600 = 1/3 and 300/(100 + 300) = 3/4. Thus, p = (1/2)(2/3)+(1/3)(3/4) = (1/3)+(1/4) = 7/12.