I often gets questions about gambling and the lottery. Just the other day I was asked to figure the odds of a tossed coin landing on 'heads' after 9 'heads' were tossed in a row.
I explained and the person still does not believe the answer. I told him that if a fair coin is tossed and heads comes up 9 times in a row it has no effect on the next toss. A coin has no memory so there is still the usual 50-50 chance of tails or heads. The previous 9 or 900 tosses have nothing to do with that next toss.
The confusion is about the probability of tossing 10 'heads' in a row. That's a whole 'nother smoke.
The probability of one head in a row is 1 out of 2 (.5), the probability of 2 heads in a row is 1 out of 4 (.5 x .5), 3 heads is 1 out of 8 (.5 x .5 x .5) and so on. So the probability of 10 heads in a row is 1 out of 1024 (.5 to the 10th power) or .0009765625.
In a real world case where someone tosses 9 heads in a row I would be concerned about the fairness of the coin.