My friends had the opportunity to trick me since we regularly played Risk (and I would have been highly amused if they had done so). Since the dice were mine and were distinctive they would have had to get trick dice that matched my own. Then they would have had to wait for the right game opportunity, e.g., my 26 armies against my opponent’s last remaining army on his last territory. Knowing my friends very well, it doesn’t seem likely to me that they would go to all that trouble and then never laugh about how they fooled me.
My friends didn’t appear all that surprised by the event. Both believed in “luck” and neither had a mathematical understanding of just how rare such a “chance” event would be. I interacted on a daily basis with these friends for several more years and they consistently expressed the view that it had been a “lucky run”, unusual but nothing earth shaking. My impression was that they viewed it as a one in a thousand event consistent with their belief in lucky people and lucky streaks. To me it was amazing because I didn’t believe in “lucky people” and could calculate how unlikely such an event was. (“Rare” events might happen frequently and pass relatively unnoticed because people just can’t calculate how unlikely the events really are.)
I have difficulty believing that trick dice would work well enough to fool me in this particular case. My opponent didn’t roll a string of sixes. He beat me with sixes, fives, fours, threes, and even a two. (The two sticks in my mind because at the time I thought to myself that I seemed to be trying to lose.) We are talking about a trick die that occasionally rolled every number except a 1 but still managed to beat or tie my best die for 25 times in a row. That is unbelievable control of little plastic cubes considering we are rolling at the same time using die cups.
I have no explanation for that event. I never saw my friend do anything similar before or after and I really don’t think he had anything to do with it. In my opinion the three of us were observers in something strange but none of us were really in control. I don’t attribute it to luck or psychic powers.
PS. If I were reading some anonymous poster describing this event on the Internet, I’d assume he was lying, was delusional, had been tricked, or was badly mis-remembering the event. However, people who have personally experienced something similar might get something out of my description.
So I’m not a mathematician but we note the outcomes of chance events all the time probably thousands to tens of thousands of times in your life depending on how much gaming you do. Given about 1000 low-likelihood events per person over their lifetime (which I’m basically making up, but I think its conservative) 1 in 100 million should experience 1 in 100 billion events, right? So basically there might be two other people with stories like yours living in the US. It is definitely a neat story, but I don’t think its the kind of thing we should never have expected to happen. Its not like the quantum tunneling of macroscopic objects or anything.
1000 is extremely conservative. Every time you play any game with an element of chance—risk, backgammon, poker, scrabble, blackjack, or even just flipping a coin—the odds against you getting the exact sequence of outcomes you do get will be astronomical. So the limiting factor on how many unbelievable outcomes you perceive in a lifetime is how good you are at recognizing patterns as “unusual”. Somebody who studied numerology or had “lucky numbers” or paid attention to “lucky streaks” would see them all the time.
In the case at hand, that same series of rolls would be just as unlikely if it had happened at the beginning of the game or in the middle or spread throughout the match and hadn’t determined the outcome. Unless there was something special about this particular game that made its outcome matter—perhaps it was being televised, there was a million dollars bet on it, or it was otherwise your last chance to achieve some important outcome—the main thing that makes that sequence of rolls more noteworthy than any other sequence of rolls of equivalent length is selection bias, not degree of unlikeliness.
“the odds against you getting the exact sequence of outcomes you do get will be astronomical”
People notice and remember things they care about. Usually people care whether they win or lose, not the exact sequence of moves that produced the result. For an event to register as unusual a person must care about the outcome and recognize that the outcome is rare. The Risk game was special because I cared enough about the outcome to notice that I was losing, because the outcome (of losing) with 26 vs. 1 armies was incredibly unlikely, and because I could calculate the odds against such an outcome occurring due to chance.
During an eighth grade science class in Oklahoma, my older sister was watching as her teacher gave a slide presentation of his former job as a forest ranger. One of the first slides was a picture of the Yellow Stone National Park entrance sign. Four young children were climbing on the sign and parked next to the sign was a green Ford Mercury. My sister jumped out of her chair yelling, “That’s us.” Sure enough that picture had captured a chance encounter years ago, far away, before my sister and her teacher had ever met. (A couple of years later I took the same class and saw the same slide. I would never have noticed our family climbing on that sign if I hadn’t remembered my sister describing her classroom experience at the dinner table.)
So very unlikely events do occur. However people are seldom in a position to both notice the event and calculate just how rare the event really is.
“So basically there might be two other people with stories like yours living in the US.”
Yes. The event has significance to me only because it happened to me. I would significantly discount the event if I heard about it second hand.
The event has significance to me only because it happened to me. I would significantly discount the event if I heard about it second hand.
Why in the world should who the event happens to make a difference? This is anthropic bias. The fact if these things happen at all they’re going to happen to someone. That fact that it was you isn’t significant in any way.
“Why in the world should who the event happens to make a difference?”
I question the surface view of the world and the universe. E.g., I wouldn’t be greatly surprised to discover that “I” am a character in a game. To the extent that I understand reality, my “evidence model” is centered on myself and diminishes as the distance from that center increases.
In the center I have my own memories combined with my direct sensory perception of my immediate environment. I also have my internal mental model of myself. This model helps me evaluate the reliability of my memories and thoughts. E.g., I know that my memory is less consistent than information that I store on my computer and then directly access with my senses. I also observe myself making typing errors, spelling errors, and reasoning errors. Hence, I only moderately trust what my own mind thinks and recalls. (On science topics my internal beliefs are fairly consistent with information I receive from outside myself. On religious and political topics, not so much.)
Friends, family, and co-workers fill the next ring. I would treat second hand evidence from them as slightly less reliable and slightly less meaningful. Next would be friends of friends. Then US citizens. Then humans. The importance I place on events and evidence decreases as my connection to the person decreases. Some humans are in small, important sets, while others are in very large, unimportant sets. That some human won the lottery isn’t unusual. That I won the lottery is. Of course to some guy in India, my winning the lottery wouldn’t be special because he has no special connection to me.
If I won a 1-in-100 million lottery I would adjust my beliefs as to the nature of reality somewhat. I would decrease my belief that reality is mundane and increase my belief that reality is strange.
When you say “that is unbelievable control”, you seem to be assuming the exact outcome with trick dice would be exactly and entirely predetermined. But there’s no reason to think that. The trick dice would only have to make winning much more likely to pull your “impossible” odds down into the realm of the possible. What you describe as a die that “occasionally rolled every number except a 1” is what you would expect to see if the “1″ side were weighted a bit—it would often roll a 6, sometimes roll a number adjacent to 6, and never roll 1. Contrariwise, it’s possible that the three dice facing it could have been rigged to do poorly. If a die with the “1” side weighted faced three dice with the 6 side weighted, that could do the trick.
Some amount of dice rigging could make your loss expected or reasonably likely but not guaranteed. And yes, it’s unlikely your friend would (a) weight your dice, (b) waste this ability on a meaningless game of risk, and (c) keep up the act all these years, but it’s not 1-in-100-billion unlikely. People playing little tricks or experimenting on their friends is something that does happen in the world as we know it, therefore it could have happened to you.
″...it would often roll a 6, sometimes roll a number adjacent to 6...”
Assuming standard probability applied to my three dice, the odds of my rolling at least one 6 are 1 - (5/6)^3 or approximately 0.4. Assume that the “trick” die rolls a 6 half the time. (Remember I was watching as my opponent also rolled 5′s, 4′s, and 3′s.) Then the probability that I would win a battle is at least 0.4 x 0.5 = 0.2. The attacker odds are actually higher since the attacker would usually win if the defender rolls anything but a 6. My estimate is that with the trick die, the defender would win with frequency around 0.6. So the probability that the defender would win 24 battles is around 0.6^24 or about 1-in-100,000.
“And yes, it’s unlikely your friend would (a) weight your dice, (b) waste this ability on a meaningless game of risk, and (c) keep up the act all these years, but it’s not 1-in-100-billion unlikely.”
There is also (d), even with a “trick” die the event would only be expected to happen with frequency 1-in-100,000. Now combine that low probability with the low probabilities of (a), (b), and (c) also being true. I agree that it is more likely that (a), (b), (c), and (d) are all true is more likely than that a 1-in-100-billion event happened. However, I’m not claiming a 1-in-100-billion event happened. I’m claiming that it is more likely that something unknown occurred, i.e., I have no scientific explanation for the event.
Yes, you do: all four dice were weighted. You did your math assuming only one of them was weighted, but if they all were then the event you saw wasn’t unlikely at all. Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side. How strongly weighted do the dice have to be (that is, what should p be) for 26 consecutive victories for the defender are assured?
The defender automatically wins on a 5 or 6, which come up with probability p + (1-p)/4. If the defender rolls a 2, then for the defender to win, each of the attacker’s dice must either be a 1 (which it is with probability p) or a 2 (with probability (1-p)/4), so the defender wins in this case with probability (p+(1-p)/4)^3. The cases where the defender rolls a 3 or 4 are similar. Summing all the cases, we get that the defender wins with probability
To win 26 times in a row with 50% probability, the defender would have to win each battle with probability 0.974. To win 26 times in a row with 95% probability, the defender would have to win each battle with probability 0.998.
(1/64)(-9p^4-6p^3+54p+25) > .974
--> p > .841
(1/64)(-9p^4-6p^3+54p+25) > .998
--> p > .958
In other words, to produce the event you saw with 50% reliability would require weighted dice that worked 84% of the time. To produce the event you saw with 95% reliability would require weighted dice that worked 96% of the time. I’m unable to find any good statistics on the reliability of weighted dice, but 84% sounds about right.
I used three reddish, semi-transparent plastic dice with white dots (as I always did). My opponent used standard opaque, plastic ivory dice with black dots. I noticed nothing unusual about the dice and by the end of the run I was examining dice, cups, methods of rolling closely.
“Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side.”
This assumption does not match my recollection of the dice rolls. As I stated previously, I rolled 6′s, 5′s, 4′s, 3′s, 2′s, and 1′s. I also never rolled a 1,1,1 which should happen frequently if my dice were heavily weighed to roll 1′s. Nor do I remember rolling large numbers of 1′s.
Your probability model for a trick die also fails to match my observations of my opponents die rolls. E.g., in your model my opponent would be expected to roll similar numbers of 5′s, 4′s, 3′, and 2′s. However, he only rolled a 2 once and he rolled far more 5′s than 3′s.
Besides with your probability model for trick dice, I would have easily noticed if my opponent rolled a 6 84% of the time and I never rolled a 6 at all.
PS You used 26 in the above calculation. I had 26 armies and in Risk the attacker must have at least 4 armies to roll three attack dice. So the 3vs1 dice scenario only happened 23 times.
re: Magician’s Trick
My friends had the opportunity to trick me since we regularly played Risk (and I would have been highly amused if they had done so). Since the dice were mine and were distinctive they would have had to get trick dice that matched my own. Then they would have had to wait for the right game opportunity, e.g., my 26 armies against my opponent’s last remaining army on his last territory. Knowing my friends very well, it doesn’t seem likely to me that they would go to all that trouble and then never laugh about how they fooled me.
My friends didn’t appear all that surprised by the event. Both believed in “luck” and neither had a mathematical understanding of just how rare such a “chance” event would be. I interacted on a daily basis with these friends for several more years and they consistently expressed the view that it had been a “lucky run”, unusual but nothing earth shaking. My impression was that they viewed it as a one in a thousand event consistent with their belief in lucky people and lucky streaks. To me it was amazing because I didn’t believe in “lucky people” and could calculate how unlikely such an event was. (“Rare” events might happen frequently and pass relatively unnoticed because people just can’t calculate how unlikely the events really are.)
I have difficulty believing that trick dice would work well enough to fool me in this particular case. My opponent didn’t roll a string of sixes. He beat me with sixes, fives, fours, threes, and even a two. (The two sticks in my mind because at the time I thought to myself that I seemed to be trying to lose.) We are talking about a trick die that occasionally rolled every number except a 1 but still managed to beat or tie my best die for 25 times in a row. That is unbelievable control of little plastic cubes considering we are rolling at the same time using die cups.
I have no explanation for that event. I never saw my friend do anything similar before or after and I really don’t think he had anything to do with it. In my opinion the three of us were observers in something strange but none of us were really in control. I don’t attribute it to luck or psychic powers.
PS. If I were reading some anonymous poster describing this event on the Internet, I’d assume he was lying, was delusional, had been tricked, or was badly mis-remembering the event. However, people who have personally experienced something similar might get something out of my description.
So I’m not a mathematician but we note the outcomes of chance events all the time probably thousands to tens of thousands of times in your life depending on how much gaming you do. Given about 1000 low-likelihood events per person over their lifetime (which I’m basically making up, but I think its conservative) 1 in 100 million should experience 1 in 100 billion events, right? So basically there might be two other people with stories like yours living in the US. It is definitely a neat story, but I don’t think its the kind of thing we should never have expected to happen. Its not like the quantum tunneling of macroscopic objects or anything.
1000 is extremely conservative. Every time you play any game with an element of chance—risk, backgammon, poker, scrabble, blackjack, or even just flipping a coin—the odds against you getting the exact sequence of outcomes you do get will be astronomical. So the limiting factor on how many unbelievable outcomes you perceive in a lifetime is how good you are at recognizing patterns as “unusual”. Somebody who studied numerology or had “lucky numbers” or paid attention to “lucky streaks” would see them all the time.
In the case at hand, that same series of rolls would be just as unlikely if it had happened at the beginning of the game or in the middle or spread throughout the match and hadn’t determined the outcome. Unless there was something special about this particular game that made its outcome matter—perhaps it was being televised, there was a million dollars bet on it, or it was otherwise your last chance to achieve some important outcome—the main thing that makes that sequence of rolls more noteworthy than any other sequence of rolls of equivalent length is selection bias, not degree of unlikeliness.
“the odds against you getting the exact sequence of outcomes you do get will be astronomical”
People notice and remember things they care about. Usually people care whether they win or lose, not the exact sequence of moves that produced the result. For an event to register as unusual a person must care about the outcome and recognize that the outcome is rare. The Risk game was special because I cared enough about the outcome to notice that I was losing, because the outcome (of losing) with 26 vs. 1 armies was incredibly unlikely, and because I could calculate the odds against such an outcome occurring due to chance.
re: Recognizing low probability events.
During an eighth grade science class in Oklahoma, my older sister was watching as her teacher gave a slide presentation of his former job as a forest ranger. One of the first slides was a picture of the Yellow Stone National Park entrance sign. Four young children were climbing on the sign and parked next to the sign was a green Ford Mercury. My sister jumped out of her chair yelling, “That’s us.” Sure enough that picture had captured a chance encounter years ago, far away, before my sister and her teacher had ever met. (A couple of years later I took the same class and saw the same slide. I would never have noticed our family climbing on that sign if I hadn’t remembered my sister describing her classroom experience at the dinner table.)
So very unlikely events do occur. However people are seldom in a position to both notice the event and calculate just how rare the event really is.
“So basically there might be two other people with stories like yours living in the US.”
Yes. The event has significance to me only because it happened to me. I would significantly discount the event if I heard about it second hand.
Why in the world should who the event happens to make a difference? This is anthropic bias. The fact if these things happen at all they’re going to happen to someone. That fact that it was you isn’t significant in any way.
“Why in the world should who the event happens to make a difference?”
I question the surface view of the world and the universe. E.g., I wouldn’t be greatly surprised to discover that “I” am a character in a game. To the extent that I understand reality, my “evidence model” is centered on myself and diminishes as the distance from that center increases.
In the center I have my own memories combined with my direct sensory perception of my immediate environment. I also have my internal mental model of myself. This model helps me evaluate the reliability of my memories and thoughts. E.g., I know that my memory is less consistent than information that I store on my computer and then directly access with my senses. I also observe myself making typing errors, spelling errors, and reasoning errors. Hence, I only moderately trust what my own mind thinks and recalls. (On science topics my internal beliefs are fairly consistent with information I receive from outside myself. On religious and political topics, not so much.)
Friends, family, and co-workers fill the next ring. I would treat second hand evidence from them as slightly less reliable and slightly less meaningful. Next would be friends of friends. Then US citizens. Then humans. The importance I place on events and evidence decreases as my connection to the person decreases. Some humans are in small, important sets, while others are in very large, unimportant sets. That some human won the lottery isn’t unusual. That I won the lottery is. Of course to some guy in India, my winning the lottery wouldn’t be special because he has no special connection to me.
If I won a 1-in-100 million lottery I would adjust my beliefs as to the nature of reality somewhat. I would decrease my belief that reality is mundane and increase my belief that reality is strange.
When you say “that is unbelievable control”, you seem to be assuming the exact outcome with trick dice would be exactly and entirely predetermined. But there’s no reason to think that. The trick dice would only have to make winning much more likely to pull your “impossible” odds down into the realm of the possible. What you describe as a die that “occasionally rolled every number except a 1” is what you would expect to see if the “1″ side were weighted a bit—it would often roll a 6, sometimes roll a number adjacent to 6, and never roll 1. Contrariwise, it’s possible that the three dice facing it could have been rigged to do poorly. If a die with the “1” side weighted faced three dice with the 6 side weighted, that could do the trick.
Some amount of dice rigging could make your loss expected or reasonably likely but not guaranteed. And yes, it’s unlikely your friend would (a) weight your dice, (b) waste this ability on a meaningless game of risk, and (c) keep up the act all these years, but it’s not 1-in-100-billion unlikely. People playing little tricks or experimenting on their friends is something that does happen in the world as we know it, therefore it could have happened to you.
Though I like Jack’s explanation too.
″...it would often roll a 6, sometimes roll a number adjacent to 6...”
Assuming standard probability applied to my three dice, the odds of my rolling at least one 6 are 1 - (5/6)^3 or approximately 0.4. Assume that the “trick” die rolls a 6 half the time. (Remember I was watching as my opponent also rolled 5′s, 4′s, and 3′s.) Then the probability that I would win a battle is at least 0.4 x 0.5 = 0.2. The attacker odds are actually higher since the attacker would usually win if the defender rolls anything but a 6. My estimate is that with the trick die, the defender would win with frequency around 0.6. So the probability that the defender would win 24 battles is around 0.6^24 or about 1-in-100,000.
“And yes, it’s unlikely your friend would (a) weight your dice, (b) waste this ability on a meaningless game of risk, and (c) keep up the act all these years, but it’s not 1-in-100-billion unlikely.”
There is also (d), even with a “trick” die the event would only be expected to happen with frequency 1-in-100,000. Now combine that low probability with the low probabilities of (a), (b), and (c) also being true. I agree that it is more likely that (a), (b), (c), and (d) are all true is more likely than that a 1-in-100-billion event happened. However, I’m not claiming a 1-in-100-billion event happened. I’m claiming that it is more likely that something unknown occurred, i.e., I have no scientific explanation for the event.
Yes, you do: all four dice were weighted. You did your math assuming only one of them was weighted, but if they all were then the event you saw wasn’t unlikely at all. Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side. How strongly weighted do the dice have to be (that is, what should p be) for 26 consecutive victories for the defender are assured?
The defender automatically wins on a 5 or 6, which come up with probability p + (1-p)/4. If the defender rolls a 2, then for the defender to win, each of the attacker’s dice must either be a 1 (which it is with probability p) or a 2 (with probability (1-p)/4), so the defender wins in this case with probability (p+(1-p)/4)^3. The cases where the defender rolls a 3 or 4 are similar. Summing all the cases, we get that the defender wins with probability
p + (1-p)/4 + (1-p)/4 * ((p+(1-p)(3/4))^3 + (p+(1-p)(2/4))^3 + (p+(1-p)(1/4))^3)
Which simplifies to
(1/64)(-9p^4-6p^3+54p+25)
To win 26 times in a row with 50% probability, the defender would have to win each battle with probability 0.974. To win 26 times in a row with 95% probability, the defender would have to win each battle with probability 0.998.
(1/64)(-9p^4-6p^3+54p+25) > .974 --> p > .841
(1/64)(-9p^4-6p^3+54p+25) > .998 --> p > .958
In other words, to produce the event you saw with 50% reliability would require weighted dice that worked 84% of the time. To produce the event you saw with 95% reliability would require weighted dice that worked 96% of the time. I’m unable to find any good statistics on the reliability of weighted dice, but 84% sounds about right.
I’m unable to find any good statistics on the reliability of weighted dice, but 84% sounds about right.
here is a set of loaded dice for sale that are advertised to roll a seven (6 on one, 1 on the other, I think) 80% of the time.
“all four dice were weighted”
I used three reddish, semi-transparent plastic dice with white dots (as I always did). My opponent used standard opaque, plastic ivory dice with black dots. I noticed nothing unusual about the dice and by the end of the run I was examining dice, cups, methods of rolling closely.
“Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side.”
This assumption does not match my recollection of the dice rolls. As I stated previously, I rolled 6′s, 5′s, 4′s, 3′s, 2′s, and 1′s. I also never rolled a 1,1,1 which should happen frequently if my dice were heavily weighed to roll 1′s. Nor do I remember rolling large numbers of 1′s.
Your probability model for a trick die also fails to match my observations of my opponents die rolls. E.g., in your model my opponent would be expected to roll similar numbers of 5′s, 4′s, 3′, and 2′s. However, he only rolled a 2 once and he rolled far more 5′s than 3′s.
Besides with your probability model for trick dice, I would have easily noticed if my opponent rolled a 6 84% of the time and I never rolled a 6 at all.
PS You used 26 in the above calculation. I had 26 armies and in Risk the attacker must have at least 4 armies to roll three attack dice. So the 3vs1 dice scenario only happened 23 times.