When i roll a die 6times,i will rarely get each number once,but if roll 600,getting 100 is more likely. Why?

Question

Why does decreasing the number of rolls make the chances of the data fitting the model decrease?

But -why- does the result get closer to the average as the sample size increases?

Answer 1

The sample size when rolling 6 times just isn't big enough. The larger the sample size, the closer to average results you will get.

But getting exactly 100 is probably pretty rare.

Answer 2

Not according to my calculations. Both of these have multinomial distributions.

If you roll a die 6 times, the probability that you get one of each of the numbers is

{6!/(1!1!1!1!1!1!)} * (1/6)^1 * (1/6)^1 * (1/6)^1 * (1/6)^1 * (1/6)^1 * (1/6)^1
= 0.015432098765

because each roll has probability 1/6, and there are 6!/1!1!1!1!1!1! = 6! ways to arrange the numbers 1, 2, 3, 4, 5, and 6.

However, a die 600 times, the probability that you get exactly 100 of each is

{600!/(100! * 100! * 100! * 100! * 100! * 100!)} * (1/6)^100 * (1/6)^100 * (1/6)^100 * (1/6)^100 * (1/6)^100 * (1/6)^100
= 2.46328583 * 10^-7
= 0.00000024632...

Again, each possibility has a 1/6 chance of coming up, and the number of ways to arrange 100 1's, 100 2's, 100 3's, 100 4's, 100 5's, and 100 6's is 600!/(100! * 100! * 100! * 100! * 100! * 100!).

If you roll 600 times, there are more possibilities, and so getting exactly any number of rolls to be anything is unlikely. However, with just 6 rolls, there is less that can possibly happen, so probabilities are larger.

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I think that your confusing your Law of Large Numbers here. If you look at the proportion of 1's say, in 6 rolls and in 600 rolls, that is a completely different story.

The proportion of 1's in 6 rolls can be quite different from 1/6. It could be 0, or 2/6, 3/6, 4/6, 5/6, or 6/6. All of these possibilities could feasibly happen. That's because, again, with so few rolls, everything hass a good possibility of happenning.

However, the proportion 1's in 600 rolls will be CLOSE to 1/6. So you wouldn't expect to get 0/600 1's or 300/600 1's or 600/600 1's, or something weird like that. 100/600 is possible, but you might get something like 101/600, or 102/600, or 99/600, and so on. However, all of these are pretty close to 1/6 This is what the Law of Large Numbers basically says.

But, you should not expect to get exactly 1/6 in 600 rolls. That statement is not what the Law of Large Numbers is saying. That's what I think you are missing.

Answer 3

Actually, there is a probablility chart of all things. I think it is called Rawl's Theory, or something like that. It's used in economics to show that the #1 will occur more often than any other #. This is followed by the #2, #3, and so on. The reason seems to be linked that with all #'s, the odds of 1 being first (a book page, a $ amount, etc) is the greatest.

For example, the IRS uses this principle to see who is lying on taxes. If you don't have enough 1's in your tax return, then it doesn't make sense.

When rolling 2 die at the same time on the Las Vegas strip while playing craps, the # 7 came up the most. This is because the odds of this # coming up is the most generic. There are more combinations of this # than any other. However, the odds of rolling a 6 and a 1 are for some reason the best. No one knows why.

Answer 4

If you roll the die six times then the probability that one of each number coming up will be :
Roll 1
Roll 2 5/6
Roll 3 4/6
Roll 4 3/6
Roll 5 2/6
Roll 6 1/6
This is 15/30 or 50% of times that it is done.

With 600 rolls the number of 100 is the average that each number comes up over all those throws. In reality this will be close to the real average due to large number of throws.

Answer 5

That is what statistics are all about. They are a game of large numbers. Even though each side of the dice gets a chance to show up it is only in the long run that they really do. Try rolling the dice 6 million times and I bet (but can't guarantee) that every one will come up fairly.

Answer 6

The only time the model will be matched perfectly everytime is with an infinite number of throws.

As 600 is closer to infinitity than 6, the changes of matching the model is much better.

Answer 7

Weak Law of Large Numbers

(Khinchin 1929). Stated the probability that the average |(X_1+...+X_n)/n-mu|infinity.

Answer 8

ODDS ARE THE SAME. THE NUMBER OF CHANCES YOU HAVE AT THOSE ODDS IS WHATS INCREASED.

Answer 9

It's actually just totally random...