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The original form of the fallacy as I understand it says that if you walk into a casino and see two 6's has been thrown on the table. That event is unlikely (probability 1 in36) however it does not give you information as to how many dice rolls there have been in the past.

So far that seems true. Double 6 is just as likely as any other combination of two dice throws.

But when you start to apply it, there seems to be cases where it is not true.

Lets suppose you see someone flip heads 20 times in a row. As I understand it the probability of that is one in 20 million. So it seems more likely to have happened if it was attempted 20 million times than if it was only attempted once. That means we can infer how many times something has happened on the basis that it was unlikely. So that seems to imply the inverse gamblers fallacy is wrong. But the logic above seemed fine. To put it another way the 20 heads in a row is just as likely as any other combinations of heads and tails. So that implies the inverse gamblers fallacy is right. So it seems the fallacy is both wrong and right which can't be true but I can't see where my slip up is. Can anyone help?

Another way to put it would be suppose the casino pays out a prize money when the double 6 but no other number is thrown. You have just heard someone has won the prize money as a double 6 was thrown. You now are given even odds on two bets . The bet is: how many times was the dice thrown before the prize money was given out? You can bet on less than 5 times or more than 5 times. What should you bet on? it seems obvious you should bet on more than 5 times but if the inverse gamblers fallacy is right it shouldn't help. but the logic behind the inverse gamblers fallacy looks sound. Again something is wrong, but I can't see what it is.