Alas, Potato Hamster remains at large and every day our hopes grow dimmer. In the summer, we tend to live with our doors open — out front there are cats and out back there are coyotes and hawks. And I fear that even if she hasn’t fallen prey to something toothsome, she may be forever doomed to roam our heating system like the eyeless fish in Smeagol’s pond.
How dim are our hopes? Well, that depends on exactly how you interpreted last week’s puzzle, which read as follows:
There are four rooms where Potato might be, plus the heating system, plus outside, all with equal probability. If she’s outside or in the heating system, she’s toast. (We refer to this as “having an adventure.”) Every night, we put flour-coated cookie sheets in three rooms and a garbage can trap in the fourth; if she’s in a room, there’s an 80 percent chance her tracks will show up on a sheet or she’ll end up in the can. If her tracks show up on a sheet, add 20 percent to our overall chance of capturing Potato. If she’s in the can, game over. Of course, every night she’s on the move and re-randomizes her probability of being in the heating system or outside, which is also game over.
What are our daily chances of recapturing Potato?
Here’s the Discussion section (or perhaps “Results” mashed-up with “Discussion”?):
Of course every night, Potato has 4/6 or .66 probability of surviving. But catching her is another proposition entirely. She is only caught in the 1/6 room that contains the trash can, and then only 80% of the time, and so each night, she has a .16*.8=.13 chance of being captured. But that’s just the start. Each night, there is also a 2/6 or .33 probability she will venture outside or into the vents ne’er to be seen again.
And now’s when it gets especially tricky. As you remember from stats 101, probability is not summative — you can’t just add her .13 chance of nightly capture until at .13+.13+.13+.13+.13+.13+.13 you’re 91% sure to catch her. And neither does a .33 probability of venturing into the wild unknown each night mean that after three days she’s 99% gone. Add to that the slightly ambiguous influence of the added 20 percent if her footprints happen to show up on a floured cookie board and you’ve got quite the conundrum.
How I meant that darn cookie sheet thing to work is that in a night after seeing her footprints, you’d have .13+.20=.33 probability of catching Potato (add 20% to the natural 13% chance of capture). But you’re not going to see footprints every night. In fact, you’ve got 3/6*.8=.4 chance of seeing footprints. And so instead of adding 20% for footprints, you add 20% of the *chance* of seeing footprints, or .2*.4=.8 or 8% increased chance every night but the first of catching Potato due to the influence of the cookie sheets (which I should mention were dumped by labradors and tracked throughout the house by kids, creating much more trouble than their 8% was worth…) (Oh, and another parenthetical: Thom asked how the cookie sheet was supposed to provide this benefit in the first place — I was imagining that if you saw prints, you’d move the trashcan to that room.)
So put it all together:
• .13 probability each night of catching the hamster
• .33 probability each night of losing the hamster forever
• An added .8 every night but the first for the influence of floured cookie sheets
The way I like to think of it is this:
• The first night is obvious: a .13 probability of catching Potato.
• You have a .13+.8=.21 chance of catching her every subsequent night she remains (safely) at large.
• So what are the chances she’s around on night #2? It’s always .66 times the previous probability.
• So night 1 = .13; night 2=.21*.66=.14; night 3=.21*.66*.66=.09; night 4=.21*.66*.66*.66=.06; etc.
At least that’s how I *meant* it to work. I accepted a couple different interpretations of the “add 20%” rule, and the winner under those somewhat less restrictive guidelines of this week’s $50 ThinkGeek gift certificate is…Patrick!
The rest of us can use the code GEEKDAD68JH to get $10 off a $50 ThinkGeek order.