Elves counted that 80% of children were good and 20% were not, and ordered toys and lump coal accordingly?

Question

But bad Santa lost the list composed by the elves, and simply dumped toys and coals randomly.

What perecentage of children got what they deserved?

Answer 1

I feel insulted. There was nothing random about it. The elves made a mistake on listing 32% of the children in the wrong catagory, and I just had to distribute the coal and toys accordingly. Now, instead of only 68% getting the right gift, 100% do. In deciding such matters, I rely on the words of Justice Jackson (US Supreme Court), to paraphrase, I am infallible because I am final!

Answer 2

OK

Let's look at this from a different way. If Santa really screwed up and was 100% wrong on the coal that he had .......

Then

60% would still get toys when they should have
20% would get coal when they should have gotton toys
20% would get toys when they should have gotton coal.

So in the worst case, 60% of the children would still get what they should.

In the best case, if Santa was lucky and did everything right, 100% of the kids would have gotton what they should have anyway.

Best + Worse /2 = 80%

My bet is 80% got what they should have (toys or coal respectively). 10% probably got coal when they should have gotton toys. And 10% got toys when they should have gotton coal.

Oh - yes - DR D is right. My scenerio only takes into account the very best and very worst situations. The are many other situations in between that I did not take into account.

IN DR D's, you have 100% + 60% + 60% +60% + 60% = 340/5 = 68%

Sorry, I was not right - but I think the 68% is right .

Hope that helps.

Answer 3

The probability of everyone getting what they deserve will approach 0 as you get more and more children.

The math of this for 10 children (the children are non-distinguishable):

Distributing 8 presents will automatically guarantee the placement of the 2 lumps of coal. The probability of this is (8*7*6*5*4*3*2*1)/(10*9*8*7*6*5*4*3) (or 8!/10P8). Alternately, you can calculate the placement of the two lumps of coal, which will give you the same answer, which is 1/45 or 2.2%.

For 100 children (just extrapolate the math), the probability is 1.86e-19 %

As you have more children, you have more places to distribute presents appropriately.

Answer 4

G - event that you choose a good child
D - event that they get what they deserve

P(G) = 0.8
P(D | G) = 0.8
P(G n D) = P(D | G) * P(G) = 0.64

P(G') = 0.2
P(D | G') = 0.2
P(G' n D) = 0.04

So P(D) = P(G n D) + P(G' n D) = 0.68
68% get what they deserve.

NOTE that this result does not depend on a large number of children. Consider the case of 5 children (4 good, 1 bad) and 5 gifts. Let's line up the children in order.
GGGGB
There are 5 equally possible ways to order the gifts to each corresponding child:
GGGGB
GGGBG
GGBGG
GBGGG
BGGGG

In 1 way, 5 children get what they deserve; and in 4 ways, 3 get what they deserve.
E(number) = 1/5 * 5 + 4/5 * 3 = 17/5
E(fraction) = 17/25 = 0.68