The High Roller Society

Quote from MusicScout >>

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I'm saying that the dice roller appears to have a weakness for generating truly random results.  I think it has a procedural reason for not generating a string of top numbers or a string of  bottom numbers.  But I don't have enough data for that yet.  If we keep going on this, I may have enough later.  As a matter of fact, I could do an analysis of what we have to see how often we hit the TN in less than the predicted 50% mark against more than the 50% mark.  I know 3d10 went very long, but I think the number immediately before that one went short.

Basically, when we roll lots of dice, my theory is we will see that the number of rolls goes over 50% more often.  The error is probably not noticeable for 2 dice, but the higher we go the more I believe we will notice.  But this could all be a coincidence, so I have to crunch some numbers.

 Interesting. Also, I'm glad you're doing this and not me. I love math - except for probability.

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I do wonder what would cause the bias. Granted I'm not an expert on all the nitty-gritty, but I'd think the in-built random functions of whichever programming language they used for the dice roller would be relatively balanced.

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So let's look at the first challenge, rolling a 4 on 1d4.

You will not roll a 4 75% of the time in one roll, so you are more likely to roll more than once.

You will not roll a 4 in the first two rolls 56.25% of the time, so you are more likely to roll more than twice.

You will not roll a four in the first three rolls 42.19% of the time, so you are more likely to roll a 4 in your first three rolls.

You will not roll a 4 on your first two rolls, and then roll a four 14.06% of the time.

This is the outcome we had in this challenge.  We hit the TN at the roll when we crossed over from less than 50% chance to more than 50% (the inverse of the numbers I gave you above).

But it isn't easy to see any conclusion because this is an extremely small sample set.  Consider also the chances of hitting the TN on a specific round.

If you say you will hit it on round 1, you will be right 25% of the time.

If you say you will hit it on round 2 (which assumes you didn't hit it on round 1), you will be right 18.75% of the time.

If you say you will hit in on round 3 (which assumes ...), you will be right 14.06% of the time.

If you say you will hit it on round 4, you will be right 10.55% of the time.

If you say you will hit it on round 5, you will be right 7.91% of the time.

And so on.

So in Case 1 we had a 56.25% chance of not rolling a 4 before round 3, and we had a 42.19% chance of rolling it before round 4.  The actual result was 3 rounds.

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So the next one was the 1d6 ...

To hit a 6 in a single roll, the chance is 16.67% and the chance to have to go further is 83.33%

The chance to hit on your second roll is 13.89% and the chance to have to go further is 69.44%.

The chance to hit on your third roll is 11.57% and the chance to go further is 57.87%.

The chance to hit on your fourth roll is 9.65% and the chance to have to go further is 48.23%.

Therefore the TN of 6 should be hit in four rolls or less.  But you still have a significant chance of going deeper than 4 rolls.  In this case we hit the answer in 4 rolls.

Going one round deeper, the chance of rolling the TN on the fifth roll is 8.04% and the chance to go further is 40.19%.

So we had a 51.77% chance to hit the TN=6 in four rolls or less and a 48.23% chance to hit it in more than four rolls.  This is what happened.

But statisticians prefer to look at the "normal result" and compute the spread from there.

So we know the "normal result" is 4 rolls.  To go one less than that is 9.65% chance.  How many rolls past 4 would it take to reach ~10%?  Well, we already know that going one more is 8.04%, so two more would be much greater.

So statisticians prefer to answer something like you have a 29.26% chance to hit the TN=6 with 3, 4 or 5 rolls.  Well, where is the 50% spread?

Well, to go more than 1 roll is 83.33%.  50% - 16.67% = 33%.  That 33% goes out to just after the sixth roll.  So we could say that there is a 50% probability that we will hit the TN=6 between rolls 2 - 6.  There is less than a 50% chance for us to hit the TN outside that range.

So if I look for the 50% result and then express a range from 25% to 75% we can see if there is any significance to the outcome of the rolls we take now that we are in the higher numbers.  Fortunately, computing the 25%, 50% and 75% thresholds is pretty easy.

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You're all making my head hurt.

It's only hard if you haven't done it before.

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Now let's jump ahead and look at this problem, hitting a 20 with 5d4 ...

The 25% result is 1419, that is there is only a 25% chance of having to roll more tan this to hit the TN.

The 50% result is 709, that is the exact point where it becomes more probable that we should have hit the number than not hit the number.

And the 75% result is 294, which is the number of times we should expect to roll to have a 25% chance of hitting the TN.

So we have a 50% chance of hitting the TN between roll #294 and roll #1419.  So far we have rolled 1248 times against 5d4 = 20.

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So I wrote down the earlier results.  I hope to have a full table of probable band width for each of them a little later today.  Some of them are crazy long and some a crazy short, like 3d8 happened in 133 rolls.  2d20 before that happened in 813 rolls and 3d10 happened in 2623 rolls.

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Will we ever win?

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Never.

Didn't think so.

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Even a blind squirrel finds a nut now and then.  Don't lose hope.