Odds of rolling 20 same each time?

[piquwee]

Hello all, table dispute...
I say there is a 1 in 20 chance to roll a 20. Even if you roll a hundred times, it will always be 1 in 20 each time.

On the other side of the table, they believe the odds of rolling multiple 20's consecutively increases the odds.

What math person out there would like to show off and answer this one?

[UnArcaneElection]

The chance of rolling a 20 is always the same, as long as the die is not wearing out significantly or accumulating dirt or thrown on a different surface each time. On the other hand, if the die is already worn out (especially if you can see that the edges are rounded and/or chipped), the chances of rolling a 20 might be significantly different from 1 in 20.

[Corathonv2]

Lady Luck has no memory. If the die is a fair one it has a 5% chance to come up "20", no matter what was rolled before. However, if the die rolls many twenties in a small number of tries then it might not be a fair die (i.e. the odds of rolling a 20 on the die might be greater than 5%).

[Voss]

You might be misunderstanding them. Yes, each individual roll is 1 in 20. But the odds of rolling 3 or 5 (or whatever) 20s in a row (multiple 20s consecutively) is not 1 in 20. You're calculating different things.

[RealAlchemy]

As others have said, the overall chances of rolling a 20 are the same every time with a properly balanced die. However, the odds of rolling 100 20s in a row are a lot lower than the odds of rolling 5 20s in a row.

[Nullpunkt]

Do you mean the odds of rolling several 20s in a row or the odds of a 20 coming up in X d20 rolls?

If what you are asking is "I will roll this d20 5 times, how likely is it that I roll 5 20s in a row?":

The odds are 1/20 * 1/20 * 1/20 * 1/20 * 1/20 = 0.0000313 % because each event (resutl is 20) has a probability of 1/20.

If you are asking "If I roll 5 times, how likely is it that one of the rolls is a 20?":

You start out by calculating the odds that you do NOT roll a 20 (an event with a probability of 19 out of 20) five times in a row:

19/20 * 19/20 * 19/20 * 19/20 * 19/20 = 77.4 %.

If the chances of NOT rolling a 20 with those 5 rolls is 77.4 %, the opposite event (so rolling a 20) has a probability of 100 % - 77.4 % = 22.6 %.

I hope that helps.

[thejeff]

RealAlchemy wrote:

As others have said, the overall chances of rolling a 20 are the same every time with a properly balanced die. However, the odds of rolling 100 20s in a row are a lot lower than the odds of rolling 5 20s in a row.

Even with a biased die, the odds of rolling a 20 will be independent of previous rolls. They may be 1 in 10 instead of 1 in 20, but they'll be 1 in 10 whether the last roll was a 1 or a 20.

And honestly the odds of rolling 100 20s in a row aren't a lot less than the odds of rolling 5. Either is pretty good evidence there's cheating going on. :)
I mean, it's possible you hit the one in 3.2 million chance, but I'd be checking the die.

[quibblemuch]

Why 20?

If this were true, it would be true for any given possible roll N. Which would mean that all dice would eventually converge on rolling some number between 1 and 20 and only rolling that number, as the probability of consecutive rolls increase.

If we lived in that kind of universe, I bet game stores would have a secret box behind the counter labeled "20" where they stored dice that had rolled enough 20s for the probability of consecutive 20s to equal 1 and they'd sell you one of those for in-store games, if you knew the passphrase...

[Sissyl]

20 in one roll is 1/20. 2 20 in two rolls is 1/20*1/20=1/400. And so on.

[RealAlchemy]

thejeff wrote:

RealAlchemy wrote:

As others have said, the overall chances of rolling a 20 are the same every time with a properly balanced die. However, the odds of rolling 100 20s in a row are a lot lower than the odds of rolling 5 20s in a row.

Even with a biased die, the odds of rolling a 20 will be independent of previous rolls. They may be 1 in 10 instead of 1 in 20, but they'll be 1 in 10 whether the last roll was a 1 or a 20.

And honestly the odds of rolling 100 20s in a row aren't a lot less than the odds of rolling 5. Either is pretty good evidence there's cheating going on. :)
I mean, it's possible you hit the one in 3.2 million chance, but I'd be checking the die.

I've seen 3 or 4 in a row, because statistical oddities can occur on occasion. Of course, I've seen 3 or 4 ones in a row as well :)

[thejeff]

RealAlchemy wrote:

thejeff wrote:

RealAlchemy wrote:

As others have said, the overall chances of rolling a 20 are the same every time with a properly balanced die. However, the odds of rolling 100 20s in a row are a lot lower than the odds of rolling 5 20s in a row.

Even with a biased die, the odds of rolling a 20 will be independent of previous rolls. They may be 1 in 10 instead of 1 in 20, but they'll be 1 in 10 whether the last roll was a 1 or a 20.

And honestly the odds of rolling 100 20s in a row aren't a lot less than the odds of rolling 5. Either is pretty good evidence there's cheating going on. :)
I mean, it's possible you hit the one in 3.2 million chance, but I'd be checking the die.

I've seen 3 or 4 in a row, because statistical oddities can occur on occasion. Of course, I've seen 3 or 4 ones in a row as well :)

Remember that 4 in a row is 20 times as likely as 5 in row. 5 is possible of course, but I'd still be checking the die. :)

[UnArcaneElection]

^Or check the player, although I've never had any luck spin-stabilizing d20s (this definitely works for d6 and d8 and sort of works for d12, and is harder than you would think for tetrahedral d4 because it is so sharp-angled, making it tend to catch, but can still be done; d20 is too rounded, although my d20 definitely has worn/chipped edges).

[Matthew Downie]

Voss wrote:

You might be misunderstanding them. Yes, each individual roll is 1 in 20. But the odds of rolling 3 or 5 (or whatever) 20s in a row (multiple 20s consecutively) is not 1 in 20. You're calculating different things.

I'd guess he's understanding them right and they're people who don't understand probability. It's pretty common.

If you roll a d20 three times and you need three twenties to succeed, your chances of success are one in 8000. But if you've rolled the first two dice and they were both twenties, the chances of success are now one in twenty.

People tend to confuse the two situations. "I just rolled two twenties in a row. The chances of me getting a third one now must be astronomically low." It's one of the classic gambler fallacies. It causes people to go to roulette tables and wait until there have been six blacks in a row, then bet on red, on the grounds that seven blacks in a row sounds unlikely. (This system has been widely tested. It doesn't work.)

[BigDTBone]

The odds of rolling two 20's in-a-row before the first die is thrown is 1 in 400. The odds of rolling a second 20 after already rolling a 20 are 1 in 20.

Rolling any two called numbers in order on a d20 will always be 1 in 400. But we just never take note of the "specialness" of rolling a 7 and a 5 in order.

[JTD]

What the Aboleth know that humans don't: It's either/or. Either you roll a particular number or you don't.

Silly humans!

[Haladir]

Every roll is an independent event.

Your chance of rolling a '20' on a d20 is 1 in 20 (5%).

Before you start rolling any dice, your chance to roll 4 20s in a row are (1/20)^4 = 1/160,000 = 0.000625%

But if you happen to have already rolled three 20s in a row, your chance on that 4th roll is still 1 in 20.

[The Sideromancer]

Something I've noticed for the d20: it's harder to productively unbalance. The standard d6 arrangement has 4,5, and 6 all one one corner, so you could just weight the opposite one. Furthermore, the coarseness (only 6 options) means you could do pretty well by weighting the side straight opposite 6. On a d20, you have finer resolution, and the high numbers are more spread out. If you weight the die for 20s, you've also significantly increased the probability of 2s.

The exception is d20s made for tracking points rather than rolling.

[Nullpunkt]

Is there a standard arrangement for the numbers on a dice? I thought the only "rule" is that the sum of opposite sides is constant (e.g., 7 on a d6, 9 on a d8 and so on).

[The Sideromancer]

Nullpunkt wrote:

Is there a standard arrangement for the numbers on a dice? I thought the only "rule" is that the sum of opposite sides is constant (e.g., 7 on a d6, 9 on a d8 and so on).

To my knowledge. For a d20, the next number in sequnec is opposite an adjacent number (example: 1 is opposite 20, 20 is next to 2, 1 is next to 19, 19 is opposite 2)