Dice math.

[Altamont Ravenard]

Technically, you perform the sum of all the values multiplied by their individual probability. In most cases, this is a bit complex, but with dice you can simplify a lot.

Since the probability for any given value of a die is 1/N where N is the number of faces, the formula for the average ends up being the sum of all values divided by N.

However, the values on a normal die generally just go from 1 to N. This makes it very easy to compute the sum as N*(N+1)/2.

Which leaves the formula as (N+1)/2, or N + 0.5. That's about all.

This gets my seal of approval

AR

 

[MerakSpielman]

This gets my seal of approval

AR

It shouldn't. His final line should read (corrected):

Which leaves the formula as (N+1)/2, or (N/2)+0.5. That's about all.

 

[Altamont Ravenard]

OK, since we're having fun with probablilty, who (besides me) can answer these:

Question 1:
You have a fair coin that is flipped 4 times, resulting in heads or tails.
Possible results:
All heads
All Tails
Three heads and 1 tail
Three tails and 1 head
2 heads and 2 tails

Which combination(s) are most likely to come up?

Question 2:
What famous mathmatical diagram can be used to predict probabilities for problems such as this? (don't know if I'm phrasing this question properly, but if you know the answer, you should recognize it)

Question 1:
16 possible combinations (2^4)
All heads = All tails = 1/16
1 head 3 tails = 1 tail 3 heads = 4/16
2 heads 2 tails = 6/16

Question 2:Isn't it the Pascal Triangle?

AR

 

[orsal]

That's true about the Central Limit Theorem only if by "closer to the average" you mean "proportionally closer to the average". If you're talking about absolute difference, the more dice you roll the farther from the average you'll generally be. If you roll four times as many dice, you'll generally get results about twice as far from the mean, but since the totals will be four times as high, twice as far (in absolute terms) is only about half as far (proportionally).

To clarify: If you roll 10d6, the standard deviation is about 5.4. The standard deviation is a convenient measuring stick for about how far from the "expected" result (35 in this case) you typically get. As a rule of thumb, about 2/3 of the time you will get a result within one standard deviation of the "expected value" (a misleading technical term). Thus about 2/3 of your 10d6 rolls will be in the range 30-40.

Now suppose you roll 20d6. The expected value is now 70, and the standard deviation is about 7.6. About 2/3 of your totals will be between 63 and 77.

How do you measure how far you are from that theoretical average? If you measure it in absolute terms, 7.6 is higher than 5.4, so with more dice you'll tend to be farther from the average. But if you measure it in relative terms, a range of 30-40 is comparable to a range of 60-80, so you get less variation with more dice.

In general, if you roll twice as many dice, multiply the standard deviation by the square root of two, which is greater than one (hence the larger absolute variation), but less than two (hence the smaller relative variation). If you roll four times as many dice, multiply the standard deviation by two (the square root of four). Nine times as many dice, multiply standard deviation by three, and so forth.

 

[Altamont Ravenard]

OK, since we're having fun with probablilty, who (besides me) can answer these:

Question 1:
You have a fair coin that is flipped 4 times, resulting in heads or tails.
Possible results:
All heads
All Tails
Three heads and 1 tail
Three tails and 1 head
2 heads and 2 tails

Which combination(s) are most likely to come up?

Question 2:
What famous mathmatical diagram can be used to predict probabilities for problems such as this? (don't know if I'm phrasing this question properly, but if you know the answer, you should recognize it)

Question 1:
16 possible combinations (2^4)
All heads = All tails = 1/16 each
1 head 3 tails = 1 tail 3 heads = 4/16 each
2 heads 2 tails = 6/16

Question 2:Isn't it the Pascal Triangle?

AR

 

[MerakSpielman]

Question 1:
16 possible combinations (2^4)
All heads = All tails = 1/16
1 head 3 tails = 1 tail 3 heads = 4/16
2 heads 2 tails = 6/16

Question 2:Isn't it the Pascal Triangle?

AR

1: Correct. Note that it's more likely to result in a 3-1 combination (8/16 - that's 50% of the time) than it is to result in a 2-2 combination.

Out of 16, the chances are:
All Heads: 1
1 head: 4
two heads: 6
1 tail: 4
All tails: 1

This corresponds to line 5 of Pascals Triangle (the line you consult increases as the number of "flips" increases):

 

[Altamont Ravenard]

WOOT!

3 years of college math and I can answer random message board probability (combinatory) trivia!

AR

 

[MerakSpielman]

WOOT!

3 years of college math and I can answer random message board probability (combinatory) trivia!

AR

Huh. I took 2 college math courses, to meet the minimum requirement. They taught this in "The Nature of Mathmatics," or, as we called it, "The Course you Take if You Don't Really Want to Know How to do Math but Still Want to Watch the Teacher do Cool Math Tricks."

 

[Olgar Shiverstone]

Here's a classification question for you:

You've flipped a coin 19 times, and each time it has come up Heads. What would you bet is the result of the 20th flip?

Spoiler

If you answered Tails, you're a gambler.
If you answered "equal probability heads or tails", you're a reasonable mathematician/engineer.
If you answered heads, you're a statistician.

 

[Numion]

It isn't really that much use however, since dice don't have memory of what they last rolled or any knowledge of statistics and so it's entirely possible (however unlikely) for your D20 to roll 1 all its life (which no doubt would be very short as you are likely to microwave it or slam it in a vice as an example to your other dice to buck their ideas up).

Actually it is very useful. When you throw several dice at once you're quite likely to get a result relatively close to the dice average multiplied the number of the dice. Most of the time 10d6 yields a result around 35, but almost never 10 or 60.

But you're right that dice have no memory.