Kittens!
Warning: this post contains probabilities - if that was you I just heard screeching, "ick," please avert your eyes and take this remedial science course.Three little kittens! Our nursing foster cat, Sadie, has been so eager to have us handle and enjoy them, so of course the inevitable happened: my wife and daughter started to guess their sex at only three days old (the kittens, that is, were at three... oh, never mind!)
They think it's two girls; one boy. They keep trying to show me why; I squint, adjust the reading glasses, and after searching for long enough to elicit a frustrated sigh, say something like, "cool!" (Which actually, translated, means, "duh - I don't get it!"). How can they see the machinery in something that tiny and fuzzy???
I'm sure they're right, but for fun let's assume that my "duh" is well-founded and that the probability of each kitten's sex is completely random - 50/50 - as far as we can tell right now. What are the chances my daughter will be correct after all? Easy one, right? Each kitten has a 50% probability of being the sex assumed; the likelihood of success for each one is an independent variable; so success is 1/2 * 1/2 * 1/2 = 1/8.
Question 1:
But we'll open it up a bit (now we get to class 3 of stats 101): what is the probability that, of the three, two will be girls and one a boy - it doesn't matter which kitten is the onion in the petunia patch. (1b:) For that matter, what is the probability that we'll have two of one sex, and one of the other?
That's still awfully easy for the stats student, but here's where it gets fun: suppose my daughter guessed that we'd have two of one sex, one of the other; while I insisted they'd be either all boys or all girls. It gets ugly; we make a bet. On the day of reckoning, with friends and family and a local film crew (hey, it has to be big, right?), she pulls out a kitten and shows the world which kind it is. I, being on film now, don't want to give up my bet one moment sooner than necessary. So even if I'm wrong, I look in the litter, and if possible pull out another kitten of the same sex.
Question 2:
Now, assuming that all that just happened, what is the probability that the final kitten is of the same sex?
Was that easy or hard? If you're really on a roll, try it the other way.
Question 3: the bet is reversed with me saying there'll be two of one sex, one of another; I'm still a competitive jerk (just for illustration purposes, mind you - hey, stop that laughing!). So when I reach in to find a kitten I'll try to pull out one of the opposite sex from what she pulled, if possible. If I'm able to do that, the bet's resolved, but what are the chances that the third and final kitten matches my daughter's kitten's sex?
So, I'd love to hear guesses and thoughts!
Hint: A portion of this scenario is actually an isomorph of another scenario that rocked the statistics community for quite a while (Question 4: what scenario was that?)
If there is any interest (and the correct answer doesn't appear), I'll post the answers next week.
posted by LetUsRun @ 7:11 AM
4 comments
4 Comments:
When blogging starts to sound like work, I get scared...
I don't wanna take all the fun at once, so I'll just poke at #1a & b.
(a) The chances are 1/4 that you'll have two of a specific sex and one of the other.
(b) It's even money that you'll have 2 of of either sex and 1 of the other.
When blogging starts to sound like work, I get scared...
:)
Hehe! And hidden behind the innocuous name of "Kittens!" Thanks for joining in!
Since no one else is biting, I'll play along a little more.
I sat staring at the screen blankly for 20min reading and re-reading question #3... If I understand it correctly, you are now in the position of betting on there being two of 'X' and one of 'Y' kittens while your daughter thinks they'll all be one or the other.
So for example, she pulls out a boy... You reach in with great determination and pull out a girl (ending the bet). You are then asking what are the chances that the 3rd kitten pulled will match be a boy (matching the sex your daughter pulled).
I'm not a statistician, but I did stay at a Holiday Inn Express last night. So, I'll wager that the odds the third kitten matches her first selected sex is 1/6.
The reason? Picking kittens blind from a bag gives us 8 possible distributions:
BBB - BBG - BGB - BGG
GGG - GGB - GBG - GBB
Based on your second pick, we know that two sets (BBB and GGG) are eliminated from consideration. Leaving us with four possible starting distributions for kittens in the bag:
BBG - BGB - BGG
GGB - GBG - GBB
You're trying to select one specific sequence from that list of six (XYX). So, you're chances are one in six.
At least, I think that's right.
Cool! Thanks for playing again!
I wondered if we'd hear one of the standard guesses, 1/2 or 1/4, but you went straight for the combinatorial analysis, eliminating the BBB and GGG cases.
(Actually I think there are two cases in the six patterns that meet the XYX model, which would boost the rate to 1/3)
So far so good, except for the "monkeywrench" of my choice to purposefully choose a different sex animal, if possible (thus selecting a B or G out-of-sequence in some cases).
I don't know if that was a good clue, or if it only muddies things further to my many (oops: i mean both) readers - I'd better give up the goods and let y'all know of the equivalent scenario ... it's really a fun story - tomorrow!
Cheers!
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