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Subject: Are the dice fair?
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I've now played Thunderbirds three times. In two of the games, we ended up rolling a doublehood.
That's only supposed to happen one roll in 36. Now, I wasn't keeping count but I'm pretty sure we made considerably fewer than 72 rolls between those three games.
I'm willing to believe we were just unlucky. But equally, I'm willing to believe that dice with heavily indented symbols will turn out to be somewhat weighted.
Unfortunately, I don't own either of the sets I've played, so I can't now go and roll the dice a bunch more times, but I was wondering if anyone else had had a similar experience?
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In a completely random series of tests there is no relationship between test instances. So just 'cos you roll 2 doubleHoods in a game (or even in consecutive rolls), it doesn't mean that the dice are biased. And even 2 doublehoods in 72 rolls says nothing about bias  you need a much larger sample size before you can make any determination of bias.
Or you could make a salt solution that allows the dice to float. If the dice float Hood side up on a regular basis then I'd suspect bias. (Of course, I don't know whether the ink will wash off in a salt solution, so don't blame me if it all goes horribly wrong)
By the way, did you notice rolling any double5s or double4s etc.? If you didn't then you may be experiencing observational bias. We tend to remember the bad dice rolls rather than the good ones. Partly this is because they are often more significant than good ones. Who cares  if you need say, 6  and you roll a double3, double4, double5 or just a 4+2 or 5+1? Nobody. So you don't register the doubles, but you sure remember the doubleHood.

 Last edited Mon Nov 9, 2015 1:47 pm (Total Number of Edits: 1)
 Posted Mon Nov 9, 2015 1:40 pm
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 Given that the symbols are recessed, these dice are going to be statistically significantly unfair over many rolls. But as for being so noticeably unfair over three games, I'm a counterexampleI've played a total of 5 games, and have never seen a doubleHood. And on the other hand, I've only seen one double5.
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 Take joy from your wins; take lessons from your losses.United States
Davidsonville
Maryland 
I'll add that "equal probability of all sides appearing" is a factual assessment, "fair" is a matter of opinion. Once your expectations are set for those dices' particular foibles you'll play appropriately, and they don't favor one player over another in this particular game.
S.

 Last edited Mon Nov 9, 2015 2:42 pm (Total Number of Edits: 1)
 Posted Mon Nov 9, 2015 2:19 pm
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 Gordon WatsonUnited Kingdom
Banstead
Surrey  United KingdomBeneath this mask there is an idea.....and ideas are bulletproof. 
I just rolled my two TB dice 72 times (it's a slow day)  I got 2 double hoods and 23 hoods in total. Just about bang on the probabilities, so no problems with the dice
The amount of plastic removed for the recesses is unlikely to have any noticeable skewing of the results over the number of rolls likely for an individual game  you will need to roll 1,000's of times to see any possible effect.
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Darth Binkly wrote:Given that the symbols are recessed, these dice are going to be statistically significantly unfair over many rolls. But as for being so noticeably unfair over three games, I'm a counterexampleI've played a total of 5 games, and have never seen a doubleHood. And on the other hand, I've only seen one double5.That would only be true if the missing dice material is different between faces and if the net effect is to move the centre of gravity of the die away from the physical centre of the die  and by a sufficient amount that it has an effect on the mean value rolled by the die. None of this is inherent or automatic in the fact that "the symbols are recessed"  as you assert that it is.
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Maybe we can find someone out there to do a centre of mass validation on the dice and see if there might be something "off".
I would expect that to be part of the design/manufacturing process but I guess it depends on the professionalism of the company making the dice.
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mccrispy wrote:In a completely random series of tests there is no relationship between test instances. So just 'cos you roll 2 doubleHoods in a game (or even in consecutive rolls), it doesn't mean that the dice are biased. And even 2 doublehoods in 72 rolls says nothing about bias  you need a much larger sample size before you can make any determination of bias.Yesyes. I do have a mathematical background, and do understand statistics.
I'm fully aware that so far I only have an anecdote rather than significant data. I'm also aware that even with fair dice half the people here will have experienced more doublehoods than average. (8
But still... it was enough to set me wondering.
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Sagrilarus wrote:I'll add that "equal probability of all sides appearing" is a factual assessment, "fair" is a matter of opinion.Hmm.
I'd say the players have a "right" to expect a sixsided die to roll each face with equal probability. The game's allowed to spring plenty of nasty surprises on the players, but it doesn't feel appropriate for that to be one of them.
Also, one assumes the game was prototyped, playtested and balanced using a nonproduction die. If hoods turned up for players more than they did for Matt Leacock, that's an issue.
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rmair62 wrote:Maybe we can find someone out there to do a centre of mass validation on the dice and see if there might be something "off".It might depend on how much they think it matters.
I would expect that to be part of the design/manufacturing process but I guess it depends on the professionalism of the company making the dice.
My mother's copy of Catan, for example, happens to have flagrantly biased dice. Eventually, this frustrated me so much I bought her a pair of highquality dice.
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domus_ludorum wrote:I just rolled my two TB dice 72 times (it's a slow day)  I got 2 double hoods and 23 hoods in total. Just about bang on the probabilities, so no problems with the dice ;)That's reassuring  thanks!
Next time I'm near a Pandemic set, I may make some rolls myself. (Possibly during the downtime if anyone ever insists on playing TashKalar again.)
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clivej wrote:These two statements appear to be at odds. The mere act of asking such a question also appeared to preclude such an understanding. My mistake. Apparently even statisticians can think with their "gut" when it comes to probabilitybased problems.mccrispy wrote:In a completely random series of tests there is no relationship between test instances. So just 'cos you roll 2 doubleHoods in a game (or even in consecutive rolls), it doesn't mean that the dice are biased. And even 2 doublehoods in 72 rolls says nothing about bias  you need a much larger sample size before you can make any determination of bias.Yesyes. I do have a mathematical background, and do understand statistics.
[...]
But still... it was enough to set me wondering.
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mccrispy wrote:I don't think those statements are at odds with each other. There is no reason to associate understanding of mathematical probabilities with a "gut feel" that the dice are "off" in some way.clivej wrote:Yesyes. I do have a mathematical background, and do understand statistics.These two statements appear to be at odds. The mere act of asking such a question also appeared to preclude such an understanding. My mistake. Apparently even statisticians can think with their "gut" when it comes to probabilitybased problems.
[...]
But still... it was enough to set me wondering.
Quite the opposite, in fact, your "gut" could be telling you that you should not roll that many "Hood" results while your eyes tell you different. That is what makes me wonder if there might be a manufacturing flaw that is causing The Hood to show up more often than expected.
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mccrispy wrote:Those two statements aren't at odds.clivej wrote:These two statements appear to be at odds. The mere act of asking such a question also appeared to preclude such an understanding. My mistake. Apparently even statisticians can think with their "gut" when it comes to probabilitybased problems.mccrispy wrote:In a completely random series of tests there is no relationship between test instances. So just 'cos you roll 2 doubleHoods in a game (or even in consecutive rolls), it doesn't mean that the dice are biased. And even 2 doublehoods in 72 rolls says nothing about bias  you need a much larger sample size before you can make any determination of bias.Yesyes. I do have a mathematical background, and do understand statistics.
[...]
But still... it was enough to set me wondering.
But theseQuote:I do have a mathematical background, and do understand statistics.Quote:I'm also aware that even with fair dice half the people here will have experienced more doublehoods than average.are another matter entirely.
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Psauberer wrote:Those two statements aren't at odds.Oh really? :p
But theseQuote:I do have a mathematical background, and do understand statistics.Quote:I'm also aware that even with fair dice half the people here will have experienced more doublehoods than average.are another matter entirely.
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mccrispy wrote:It seems clear to me that there is an observable difference in the amount of material removed, with the 1 as the low extreme, and probably the Hood as the high. The Hood is opposite the 1. From that, it follows that the CG is shifted. Over enough rolls, any shift will have an effect that, in the absence of other unknown biases that exactly cancel it, is statistically significant.Darth Binkly wrote:Given that the symbols are recessed, these dice are going to be statistically significantly unfair over many rolls. But as for being so noticeably unfair over three games, I'm a counterexampleI've played a total of 5 games, and have never seen a doubleHood. And on the other hand, I've only seen one double5.That would only be true if the missing dice material is different between faces and if the net effect is to move the centre of gravity of the die away from the physical centre of the die  and by a sufficient amount that it has an effect on the mean value rolled by the die. None of this is inherent or automatic in the fact that "the symbols are recessed"  as you assert that it is.
So, I do assert, without conducting the experiment and applying the law of large numbers, that there is a bias that is consequent on the recesses having different areas. To answer the OP's question, the dice are not fair. Are they fair enough? That remains to be seen, but you're correct that the double Hoods are not evidence that they're not.
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Darth Binkly wrote:Yes, I was quite careful with my weaselyworded phrasing . Note my "and by a sufficient amount that it has an effect on the mean value rolled by the die" clause. Though I should perhaps have said "noticeable effect".mccrispy wrote:It seems clear to me that there is an observable difference in the amount of material removed, with the 1 as the low extreme, and probably the Hood as the high. The Hood is opposite the 1. From that, it follows that the CG is shifted. Over enough rolls, any shift will have an effect that, in the absence of other unknown biases that exactly cancel it, is statistically significant.Darth Binkly wrote:Given that the symbols are recessed, these dice are going to be statistically significantly unfair over many rolls. But as for being so noticeably unfair over three games, I'm a counterexampleI've played a total of 5 games, and have never seen a doubleHood. And on the other hand, I've only seen one double5.That would only be true if the missing dice material is different between faces and if the net effect is to move the centre of gravity of the die away from the physical centre of the die  and by a sufficient amount that it has an effect on the mean value rolled by the die. None of this is inherent or automatic in the fact that "the symbols are recessed"  as you assert that it is.
So, I do assert, without conducting the experiment and applying the law of large numbers, that there is a bias that is consequent on the recesses having different areas. To answer the OP's question, the dice are not fair. Are they fair enough? That remains to be seen, but you're correct that the double Hoods are not evidence that they're not.
We see these sorts of engraved (moulded?) dice all to often IMO and with little attempt to infill with material the same density as the die material  or even of equivalent weight to the removed material. But whether they have a significant effect on gameplay rather than aesthetics, I really don't know.

 Last edited Mon Nov 9, 2015 5:26 pm (Total Number of Edits: 1)
 Posted Mon Nov 9, 2015 5:25 pm
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 They call me....United States
Riverside
CaliforniaSpecious arguments are not proof of trollish intent. 
clivej wrote:Yeah. Maybe you really want to think about the mode, and just give up on medians and means.Psauberer wrote:Those two statements aren't at odds.Oh really? :p
But theseQuote:I do have a mathematical background, and do understand statistics.Quote:I'm also aware that even with fair dice half the people here will have experienced more doublehoods than average.are another matter entirely.
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MisterCranky wrote:I take a group of people, give each person a pair of fair dice, then ask them all to roll their dice many times.clivej wrote:Yeah. Maybe you really want to think about the mode, and just give up on medians and means.Psauberer wrote:Those two statements aren't at odds.Oh really? :p
But theseQuote:I do have a mathematical background, and do understand statistics.Quote:I'm also aware that even with fair dice half the people here will have experienced more doublehoods than average.are another matter entirely.
Then I ask each person to report how many times in 36 they rolled a double six. The average (mean, median or mode) will be 1.
Half of them will have rolled a double six more often than that; half less often.
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Darth Binkly wrote:It seems clear to me that there is an observable difference in the amount of material removed, with the 1 as the low extreme, and probably the Hood as the high. The Hood is opposite the 1. From that, it follows that the CG is shifted.Mmm. I had the physics of the situation in mind as well as my anecdata when I asked.
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 Take joy from your wins; take lessons from your losses.United States
Davidsonville
Maryland 
clivej wrote:No die ever made provides true equal probability; most d6 provide very close to equal probability. I'd wager these do too. I know someone that has redesigned the sixsider to truly have equal mass through all axes in motion and at rest, but that's just improving on an otherwise dependable design.Sagrilarus wrote:I'll add that "equal probability of all sides appearing" is a factual assessment, "fair" is a matter of opinion.Hmm.
I'd say the players have a "right" to expect a sixsided die to roll each face with equal probability.
All that said, you take what fate gives you, and if you roll the dice 1000 times and discover a bias, you work with it. Every game has flaws, physical or logical.
S.

 Last edited Tue Nov 10, 2015 11:44 am (Total Number of Edits: 1)
 Posted Mon Nov 9, 2015 9:39 pm
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I haven't seen anything in the documentation to suggest that it's expected that The Hood will appear once in every 6 rolls of a die. He has a hard enough time as it is, cut him some slack
If there is a bias, albeit slight, it could just as well be by design.
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clivej wrote:I take a group of people, give each person a pair of fair dice, then ask them all to roll their dice many times.No. The reason for the questioning is though you assert you have a mathematical background you keep saying things that appear to indicate you have never internalised the thinking associated.
Then I ask each person to report how many times in 36 they rolled a double six. The average (mean, median or mode) will be 1.
Half of them will have rolled a double six more often than that; half less often.
In this case, if you actually take a group of people it is very unlikely the mean, median, and mode will be the same. The mode and median will almost certainly be 1, but the mean is likely to be off by a very small amount.
Your statement is wrong, its also obvious if you think about it. Some of the people will always get exactly the median and mode, in fact by definition more will get the mode than anything else. So what should happen in a fairly normal distribution is that some proportion of the people, X, get the mode/median (close to the mean), and that of the remainder (1X) half get less and half get more, but both of those amounts being less than half the people.
Your statement might have been right the first time, if by 'average' you had specifically meant the mean, and we generously included a 'most of the time' as implied, because the mean has a very good chance of being a number no test subject can achieve (such as 0.98 double hoods). But I think that is overly generous and the first respondent was right to call you on it.
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RobertBr wrote:So what should happen in a fairly normal distribution is that some proportion of the people, X, get the mode/median (close to the mean), and that of the remainder (1X) half get less and half get more, but both of those amounts being less than half the people.X tends to zero as the sample size increases.
OK, if we're being pedantic I should have covered myself with an "expected" and either said "very nearly half" or stipulated "mean" rather than the vaguer "average".
On the other hand it was a throwaway remark, and a smileyed one at that, emphasising that I knew full well many people would see more hoods than average so that wasn't necessarily remarkable. (And, as I noted, there are also physical reasons to suspect the dice might not be fair, though.)
BoardGameGeek's forums don't support nearly enough LaTeX for the level of formalism you seem to be demanding of me in order to believe that yes, honestly, I really do understand statistics, both intellectually and viscerally.
Please, go back and read again what I originally wrote. If there had been a deluge of "gosh, yes, now you mention it..." replies, it would be time to break out the lab equipment and the sigmas. As it happens, it looks like my experience was just an outlier, but I did explicitly acknowledge that possibility upfront. You seem to be suggesting it was a statistically illiterate thing even to wonder out loud about, which doesn't feel entirely fair.
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We played again this evening, and on my very first Disaster I rolled a Double Hood. I blame this thread...
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