n tHere is a more intuitive version of the same paradox.
Again, conditional on all dice rolls being even. But this time it’s either
A) 1,000,000 consecutive 6′s.
B) 999,999 consecutive 6′s followed by a (possibly non-consecutive 6).
Suppose you roll a few even numbers, followed by an extremely lucky sequence of 999,999 6′s.
From the point of view of version A, the only way to continue the sequence is a single extra 6. If you roll 4, you would need to roll a second sequence of a million 6′. And you are very unlikely to do that in the next 10 million steps. And very unlikely to go for 10 million steps without rolling an odd number.
Yes if this happened, it would add at least a million extra rolls. But the chance of that is exponentially tiny.
Whereas, for B, then it’s quite plausible to roll 26 or 46 or 2426 instead of just 6.
Another way to think about this problem is with regular expressions. Let e=even numbers. *=0 or more.
The string “e*6e*6” matches any sequence with at least two 6′s and no odd numbers.
The sequence “e*66” matches those two consecutive 6′s. And the sequence “66″ matches two consecutive 6′s with no room for extra even numbers before the first 6. This is the shortest.
Phrased this way it looks obvious. Every time you allow a gap for even numbers to hide in, an even number might be hiding in the gap, and that makes the sequence longer.
When you remove the conditional on the other numbers being even, then the “first” becomes important to making the sequence converge at all.
n tHere is a more intuitive version of the same paradox.
Again, conditional on all dice rolls being even. But this time it’s either
A) 1,000,000 consecutive 6′s.
B) 999,999 consecutive 6′s followed by a (possibly non-consecutive 6).
Suppose you roll a few even numbers, followed by an extremely lucky sequence of 999,999 6′s.
From the point of view of version A, the only way to continue the sequence is a single extra 6. If you roll 4, you would need to roll a second sequence of a million 6′. And you are very unlikely to do that in the next 10 million steps. And very unlikely to go for 10 million steps without rolling an odd number.
Yes if this happened, it would add at least a million extra rolls. But the chance of that is exponentially tiny.
Whereas, for B, then it’s quite plausible to roll 26 or 46 or 2426 instead of just 6.
Another way to think about this problem is with regular expressions. Let e=even numbers. *=0 or more.
The string “e*6e*6” matches any sequence with at least two 6′s and no odd numbers.
The sequence “e*66” matches those two consecutive 6′s. And the sequence “66″ matches two consecutive 6′s with no room for extra even numbers before the first 6. This is the shortest.
Phrased this way it looks obvious. Every time you allow a gap for even numbers to hide in, an even number might be hiding in the gap, and that makes the sequence longer.
When you remove the conditional on the other numbers being even, then the “first” becomes important to making the sequence converge at all.