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:

Hypnagogue said...

1 in 1716

Skip said...

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.

Hypnagogue said...

Oops. 1 in 26.8

Skip said...

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

Hypnagogue said...

Probability is reading comprehension with a little multiplication thrown in.

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

Daniel said...

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.

TJIC said...

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.

Don Meaker said...

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

Standard Mischief said...

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?)

reflectoscope said...

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

Jim

Anonymous said...

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

Anonymous said...

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.

Old Grouch said...

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

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

mikee said...

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.