Tuesday, November 23, 2010

What are the odds?

So I'm doing the part of the laundry that includes socks. The dryer buzzes, and I start reaching in and pulling socks out one at a time without looking.

There are five pairs of insulated winter-type socks and my two pairs of fluffy fleecy purple sock things (one striped and one polka-dotted) that I like wearing around the house in the mornings because they keep the floor from sucking the warm right out the soles of my feet, they're all happy and Dr. Seuss-looking, and they're awesome for sliding around on hardwood floors.

As I grab a sock, I lay it flat atop the washer so that when its mate comes out, I can stack them and fold them together.

Today, the washer top was too small for this chore because, in defiance of mathematical probability, I somehow managed to blindly extract one sock from each pair before getting my first duplicate. I'm tempted to try and figure the odds on that.

14 comments:

  1. The odds are actually a lot higher than you think.

    You start with 10 socks, and have a 100% chance of a non-duplicate on the first sock. On the second sock, there are 9 left, 8 of which are non-duplicates, so the chance is 8/9). On the third sock there are 8 left, 6 of which are non-duplicates, so the chance on that one is 6/8, or 3/4. 4th similarly is 4/7, and the 5th is 2/6, or 1/3. And odds multiply, so the odds of all 5 being different are:

    1*(8/9)*(6/8)*(4/7)*(2/6)

    or 384/3024, which is 8/63, which is pretty close to 1 chance in 8.

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  2. Ah - I misread that as 5 pairs, not 7. The method still works, but the answer is left as an exercise to the reader...

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  3. Probability is reading comprehension with a little multiplication thrown in.

    (I think I should get bonus points for submitting my answer by iPhone... )

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  4. Aaack. My B.A. is from a Great Books lib. arts program, but here goes: doesn't it have to be 10/10 * 4/9 * 3/8 * 2/7 * 1/6 ? (or, 1 in 126) -- at my house, the odds of one of the socks completely vanishing into the quantum vortex are significantly better.

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  5. 3.72%.

    In your first draw, there are 14 socks, and 14 of them satisfy the constraint. 14/14 that you get a valid sock.

    In your second draw, there are 13 socks, and all but one (the mate of the one you already pulled out) satisfy the constraint. 12/13 that you get a valid sock.

    In your third draw, there are 12 socks, and all but two (the mate of draw #1 and draw #2) satisfy the constraint.

    etc.

    In short:

    1: 14/14
    2: 12/13
    3: 10/12
    4: 8/11
    5: 6/10
    6: 4/9
    7 2/8

    To "win" the game, you have to win each and every round.

    Multiple 14 / 14 * 12/13 * 10/12 ...

    ...and the result is 0.0372.

    Or 3.72%.

    If you've done this 33 times, you'd expect to have a good chance of having "won" the game once by now.

    Heck, you had a 3.72% chance of winning the first time you played.

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  6. For 7 pairs of socks, prob is (14x12x10x8x6)/(14x13x12x11x10) or 48/143 for the first 5 socks with no matches.
    For 7 pairs of socks and no matchs for the first 6 socks is (14x12x10x8x6x4)/(14x13x12x11x10x9) or 64/429

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  7. Tam demonstrates the zen art of how to get people to do the algebra you forgot since high school.

    Step 1. have blog-fu

    Step 2. "bleg" without blegging.

    Step 3. profit!

    (extra credit: what is the sound of one hemisphere blegging?)

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  8. The real question is, why did the underwear gnomes set this up in the first place?

    Jim

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  9. Ya'll have too much time on your hands.Back to work.

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  10. Wait, wait, wait....
    You put seven pairs, fourteen socks in total, in the dryer and you got them ALL back?

    No MIAs?

    I think your dryer might be borked 'cuz that ain't normal.

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  11. You put seven pairs, fourteen socks in total, in the dryer and you got them ALL back?

    Hey, it IS Roseholme, not Unseen University...

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  12. I once pulled 9 sequential yellow M&Ms out of a vending machine-sized bag of that delicious candy. While studying for a quantum statistical mechanics exam at 3:00am. Spooky.

    I poured out all the rest of the bag, and it was what one would expect: several of each color, including yellow.

    Aced the exam.

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