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.

"...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.

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.