Socks

[factor-fiction]

Wildwood's query about dice and probabilities reminded me of a problem I came across recently which showed that even fairly able mathematicians can get confused by probability.

Suppose you have a sock drawer which contains just 1 pair of white socks and 1 pair of black socks. These black socks and white socks are all the same size and feel exactly the same. Now imagine that the 4 separate socks had been thrown in straight from the tumble drier and so are not paired up. Also imagine that you get dressed in the dark and pull out two socks at random. What are the chances you will pull out a pair of matching socks?

[joggerjayne]

I prefer the argument that either it will be a pair, or it won't ... so 50/50

Like the odds of winning the Lottery ... 50/50 ... either you will, or you won't!

[factor-fiction]

You just pull out 2 socks, Retrochic- no chances to try again

[joggerjayne]

Can you switch the light on?

[jim360]

The two variables aren't equally likely is the point, retrochic. As we've shown by listing the possible options, one variable (non-pair) has exactly twice as many ways or occurring as the other.

[joggerjayne]

What if you have a light on your mobile phone?

[Retrochic]

jim there are only two colours of socks white or black. You either get W/W(PAIR) B/B ( PAIR) OR W/B (NOT PAIR) That's it - I'm now with jogger jayne its 50/50 if you get a pair or not. Your equation would work if it was stipulated 'what is the chance of getting a black (or white) pair' but its not -the question is what is the chance of getting a PAIR.

[sunny-dave]

There is ONE possible white pair
There is ONE possible black pair
There are FOUR possible black and white pairs.

Therefore getting a matching pair is a 2 in 6 (= 1 in 3) chance.

Have a play with some socks and try it.

[Prudie]

You do only get a pair or not a pair so only 2 outcomes but not equal chance

[Retrochic]

sunny dave there are not four possible black and white pairs there are two....you are trying to pulling out a pair (2 together not one at a time) . W/B is totally the same as B/W so your choices are W/W OR B/B or B/W(W/B).

Factor -buy a bl88dy light bulb for your bedroom will you, you are causing havoc :-)

[jim360]

Retrochic your analysis is flawed. No other way of putting it. You need to think about how many distinct ways you can get either a pair or not. One is twice as likely as the other because the colours do matter when considering how many outcomes of drawing two socks there arr.

[jim360]

Retrochic your analysis is flawed. No other way of putting it. You need to think about how many distinct ways you can get either a pair or not. One is twice as likely as the other because the colours do matter when considering how many outcomes of drawing two socks there arr.

[factor-fiction]

Thanks for helping me out here everyone, including Retrochic for showing just how easy it is to come to an incorrect intuitive conclusion.
Now is it worth starting a new Goats and Cars thread or has that been done to death!

[sunny-dave]

I have two white socks and two black socks - how many different ways can I get a black white set?

The answer is four

White sock 1 and Black sock 1
White sock 1 and Black sock 2
White sock 2 and Black sock 1
White sock 2 and Black sock 2

These are discrete and separate pairs. Just because they are look Black and White does not make them the same physical item.

[Old_Geezer]

"What are the chances you will pull out a pair of matching socks?"

From experience probably absolute zero.

Didn't spot where it said the ones from the tumble drier were either black or white. I assumed they were patterned and blue, red, whatever.

[Retrochic]

Mioux... factor are you still miffed I beat you in correctly answering the Mensa type question yesterday with my 'intuition'? lol! ;-)

[Retrochic]

sunny-dave you are shifting the goal posts now -read the OP it says two 'pairs ' of socks so they must be the same. A 'pair' or white and a 'pair' of black.

[factor-fiction]

But what if they are shaped for left foot and right foot...

[sunny-dave]

http://static5.sdith-images.com/media/catalog/product/cache/7/thumbnail/464x/9df78eab33525d08d6e5fb8d27136e95/i/m/image_16353_1_195315_1_38903_1_48_1_270052.jpg

[factor-fiction]

Maybe subconsciously, Retrochic! Most of the grade A maths students also went with your answer so you're in good company- it's just that we have some experts in this area such as jim360 and Prudie who know the pitfalls of these types of problem and know the way to tackle them

[sunny-dave]

I will play 'pick the socks' with you for £1 a go.

Every pair comes out I give you a £1 - every non-pair comes out you give me a pound.

Assuming an honest game, I'll be about £16 up after 100 goes and will continue until you are bankrupt ...