I discovered yesterday that Labradors don’t coexist peacefully with hook-wielding four-year-olds, and so I’m crossing the fishing beach off the list of 200 possible dog outings in Boulder, Colorado. In fact, I’m crossing it off with gusto. At least for me, these things that sound like good ideas sometimes require revision in hindsight…
In any case, this week’s puzzle was a twist on the classic Birthday Problem. Here ’twas:
Imagine that each of six dogs goes out somewhere an average of once every three days. And imagine that between trails and parks and fields and out-and-abouts and algae-choked scum holes there are 200 places a dog can go in and around Boulder, all (let’s say…) with equal probability.
If it’s been exactly two years — 730 days — since Selkie’s owner picked her up from the litter, what are the chances that during this time Selkie would NOT see one of her five doggie siblings?
Though it seems kinda tractable, this turns out to be way too horribly hard for a GeekDad puzzle. Oops. Like taking labradors swimming at the fishing beach, in hindsight I think I’ll be revising the involvement needed to solve these puzzles…
Still, most entrants saw that you need to calculate the chances of Selkie not meeting another dog on any given day and then extrapolate this to her chances of not meeting another dog for 730 days.
First, there is 2/3 chance that Selkie doesn’t go out.
Then, what are the chances on the 1/3 days she goes out that she’ll meet one of her five siblings? Well, any other dog goes to a specific slot 1/3*1/200=1/600 or one of every 600 days. Effectively, there are 600 “slots” that function like days of the calendar year in the Birthday Problem, so there’s a 599/600 chance that Selkie won’t share a slot with any given dog and (599/600)^5 chance that she won’t share a slot with any of five other dogs.
The chance she won’t meet a dog on any given day is 2/3+[1/3*(599/600)^5]
Then her chances of not meeting a sibling for 730 days is [2/3+[1/3*(599/600)^5]]^730
Drum roll please. The correct answer contributed only this week by Blaine and Felicia is a 13% chance that Selkie won’t meet a sibling over two years.
Phew. I think it’s time for a word find.
Congrats to Blaine and Felicia, winners of this week’s $50 ThinkGeek gift certificate! The rest of us can use the code GEEKDAD72JL to receive a $10 discount on a ThinkGeek purchase of $50 or more.
Thanks for playing the puzzle! And don’t forget to tune in on Monday when Judd drops another installment of POTW.