Question

the local scifi club has 100 members. at a meeting, a show of hands showed that 43 members are fans of arthur c. clarke and a show of hands showed that 41 members are fans of robert a. heinlein. the random selection of a board of directors of at least how many members will guarantee that there are at least 5 clarke fans, and also at least 5 heinlein fans among the directors? if n is the right answer, you must not only explain why the random selec- tion of n members suffices, but also why selecting fewer than n members may not suffice.

Answer 1

"guarantee" !

that means have to identify the worst case scenario that mother nature and Murphy's law can throw against us to block us from our goal.

the given numbers do not provide any information about exclusivity. that means up to 41 members can be a fan of both authors.

in this case, the worst case is to pick fan after fan (100-43 = 57) of other authors, until we have 43 left.

now we pick the 2 fans that are only Clarke fans (remember, worst case), and have 41 left that are fans of both.

we would need only 3 more Clarke fans, but we don't have any heinlein fan yet. so, we need to pick 5 more to cover that.

that makes 57 + 2 + 5 = 64 picks as the minimum to guarantee that we picked at least 5 Clarke and 5 Heinlein fans.

now the other extreme : each fan is the fan of only one author.

that means we have 41+43 = 84 fans of Clarke OR heinlein.

in the worst case here, we pick one after the other that is not a fan of Clarke nor of heinlein. that is 16.

then in our worst case we only pick fans of one author.

that largest group (worst case) is 43.

now we know, that the remaining 41 are all fans of the other author.

so, we need 5 more for the guarantee.

that is then 16 + 43 + 5 = 64.

so, in both cases we need to pick at least 64 members to be sure to have at least 5 fans of Clarke and 5 fans of heinlein.

Answer 2

Selecting 5 directors ensures 5 Arthur C. Clarke and 5 Robert A. Heinlein fans, securing both groups' representation. Fewer than 5 members might lack sufficient fans, failing to ensure 5 enthusiasts of each author.

To ensure at least 5 Arthur C. Clarke and 5 Robert A. Heinlein fans among the directors, selecting 5 members is crucial. This guarantee arises from the maximum count of Clarke and Heinlein fans in the club, which is 43 and 41, respectively.

By selecting 5 members, assuming they are all Clarke and Heinlein fans, the requirement of 5 fans of each author among the directors is met. Selecting fewer members than 5 poses a risk because these individuals might not represent the true fan count proportionally.

If fewer than 5 are chosen, the chance of not selecting any Clarke or Heinlein fans increases, which fails to guarantee the needed representation. Therefore, to secure a board with at least 5 enthusiasts of both authors.