Grades 11-12 Video Solutions 2014-11&12 Video Solutions 2014 problem23

Video Transcription

Question number 23. There are nine kangaroos called greatkangs. They are either silver or gold in color. When three greatkangs meet by chance, there is a two-in-three chance that none of them is silver. How many greatkangs are gold?  Let's just work in general with, say, we have nine objects, either silver, I'll call that capital S, or gold, I'll call that capital G.  Then, choosing three, without replacement, we have the probability that the choice is G.  Equal to the number of gold objects divided by the total number.  And then the probability that the choice is G again on the second try is going to be, well, now we have eight objects left over, and we have chosen already a gold one, so it's G minus one divided by eight.  And the probability that we choose G yet again on the third pick is G minus two divided by seven.  So overall, we see that two-thirds is equal to this product, G over nine multiplied by G minus one over eight multiplied by G minus two over seven.  And that's the same thing as saying that G times G minus one times G minus two is equal to two-thirds, and then I'll multiply that by nine, by eight, by seven.  So two-thirds times nine is six, so this product is equal to eight times seven times six.  And now looking at our choices for G, the number of gold objects, or the number of gold greatkangs, we see that of the choices given, the only value of G that would fit here, this product on the right-hand side, is when G is equal to eight, G minus one would be seven, G minus two would be six.  So here G has to be equal to eight, and that is the number of greatkangs here meeting by chance with the prescribed probabilities, answer E.

Video Summary

The problem involves determining the number of gold kangaroos, "greatkangs," from a group of nine kangaroos which are either silver or gold. Given the probability condition that when three meet, there's a two-in-three chance none are silver, the constraint is translated into an equation with the number of gold kangaroos, G. By considering probabilities and solving the product equation G(G-1)(G-2) = 336 (derived from probability conditions), it's concluded that G = 8, meaning there are eight gold greatkangs.