to be utterly confused

[Confusedmyself]

In lockdown boredom we were playing scrabble earlier and have realised we can’t work out the answer to a seemingly easy problem..... AIBU to ask for your help because we are all so confused?

You have 100 scrabble tiles with a letter on one side and blank on the other, and you tip them out onto the table straight from the bag. What is the probability that exactly 50 of the tiles will be face up and 50 face down?

Please someone explain the answer because our heads are hurting!

1/100

There are options and you want the chance of one of those.

I think.

*100 options

I agree with Georgie, it seems fairly straightforward. (Except there are 101 possibilities.) Unless the OP knows otherwise?

No, it will depend on the probability of one tile landing face up. The tiles are not solid and so the heavier side will likely land on the bottom. That’s before you even think of the answer to your question!

each tile has the same probability

Now, assuming the probability for each tile landing FU or FD is equally likely, consider the options for 4 tiles.
UUUU
UUUD
UUDU
UDUU
DUUU
UUDD
UDUD
UDDU
DUUD
DUDU
DDUU
DDDU
DDUD
DUDD
UDDD
DDDD

16 possibilities but only 6 are half and half

Your second and third possibilities are the same outcome though. It doesn’t matter which 50 tiles are the right way up. They are all falling independent of each other.

The probability is about 8%.

You need three things to work out the probability of something;

1. The probability of success on a single trial (in this case, it’s 0.5 because there’s a 50/50 chance of getting the tile either facing up or facing down).

1. The number of trials (in this case, 100, because there are 100 scrabble tiles)

1. The number of successes (in this case, 50, because you want to have half of the tiles up and half down)

From that information you can work out the binomial probability. There is a formula for this, but it’s complicated to write down - I used a website which does the calculation instead: stattrek.com/online-calculator/binomial.aspx

1. They're uneven weight. A coin is 50/50. For argument's sake, let's say the chance is 75% face down (D) and 25% face up (U).

1. Each tile affects the others near it by being in contact. There is not enough time in a human lifespan to calculate all the interactions. So we'll assume that they all move completely separate from all others.

1. 100 go up. Each has a 25% of landing face up. Which is a 1 in 4 chance.

1. IIRC, you need to multiply the probability of one tile by the probability of each tile. Which is 0.25 x 0.25 x 0.25....

1. Can't you just play Snap instead?

I should have said - my calculation is based on the assumption that there is a 50/50 chance of any one tile landing face up. If this probability is different, it changes the calculation.

You can model it as a binomial distribution where n=100, r=50, p=0.5 and q=0.5
The probability of 50 facing up out of 100 tiles is therefore 0.07958923739 or ~8%

It is complicated by the fact that it's Scrabble tiles and not fair coins (it only works for spherical chickens in a vacuum)...

If you throw a fair coin, the probability of it landing heads-up is equal to the probability of it not. So the scenario of x fair coins landing heads-up is just as likely as that of x-1 or x-2 or x+i landing heads-up: that is, each of the possible scenarios has exactly the same probability, which means that said probability is 1 divided by the number of possible scenarios.

So if you have a hundred fair coins, the probability of exactly half of them landing heads-up is equal to the probability of none of them landing heads-up which is equal to the probability of all of them landing heads-up (and all possibilities in between) - ie 1/101

However, if you throw something which has, say a 2/3 chance of landing heads-up then not all of your scenarios have an equal chance.

This is where a tree diagram comes in. (I mean, you can do this for the fair-coin problem too, but when you do you'll see you don't need it.)

If you have two scrabble tiles your possibilities are

HH - 2/3 x 2/3 = 4/9
HT 2/3 x 1/3 = 2/9
TH 1/3 x 2/3 = 2/9
TT 1/3 x 1/3 = 1/9

(quick check - total adds up to one)

Here your probability of exactly half being face-up is 2/9 + 2/9 = 4/9

Add another tile and you get into 2/3 x 2/3 x 2/3 = 8/27 and so on and obviously then you can't get half of them face-up and I can't be arsed to do it for four 16/81

zscaler beat me to it!
Yes it is a binomial distribution
(That doesn't mean only two possible outcomes; it either IS something or it ISN'T so for a dice it either IS a 6 or it ISN'T a 6 (1,2,3,4 or 5) this is still "binomial"...)
For 100 things, half of which need to be what we want, assuming equal chance of it happening or not happening (Where the dice is different) it is about 8%.

My poor old calculator can't cope with 100! which is 100x99x98x97x96...

My Neave's Tables only go up to n=20

Using Excel I get 0.079589.
8% or 2 in 25.

And I think you need to ignore everything I said on the fair-coin scenario because I have fallen into the Common Fallacy (it's nearly three decades since I did this stuff...)

Well now I feel even more confused!! We had thought it was 1/101 but now I’m not so sure??!! How is it so complicated!

Ps when discussing we had assumed there was no weighting or anything like that - equal chance of it being face up or face down

Why has it been hidden @RERE21?

If you were writing it out, it would like this:

With 2 tiles, the possible outcomes are: DD / DU / UD / UU so the probability of them being half up and half down is 50/50, because 2 of the 4 possible outcomes are half up half down.

With 4 tiles, the possible outcomes are DDDD / UDDD / UUDD / UUUD / UUUU / UUDD / UUUD so the probability of getting half of them face up and half of them face down is 2/7

If you carry on until you get to 100 tiles, you can work out the probability based on what fraction of all of the possible combinations have half the tiles up and half the tiles down, then convert that to a percentage.

I love these threads because some of you get so into the maths and invested in it and I find it adorable! Ha ha!

OP, my own answer is - that will probably never happen.

7.96%

Can calculate it here...it's like 100 coin tosses and the probability of getting exactly 50 heads...

stattrek.com/online-calculator/binomial.aspx