This is because with a normal coin, each consecutive flip is done with a normal coin. after 20 heads, nothing has changed, and the 21st flip is still a normal coin.
however, once you throw a false coin into the mix, flipping 20 consecutive heads changes the condition of the 21st flip. it is now highly likely that you are holding the false coin, therefore diminishing the odds of a tails. you are right in that probability is calculated for each instance, but when new imformation is gained (ie. the higher probability of holding the fake) it factors in to the instance in question.
Having reconsidered my prior position, response answered the question "Given these 10 coins, one being double headed, what is the probability of picking one and tossing 9 heads in a row" that would be %10.17.
However, the question states, having picked one of the ten and having tossed 8 heads, what is the probability of another head.
I think pieeater has nailed this one, on the head (sorry, had to)
Wow, you just called all your responders stupid. It is obviously you who is lacking knowledge of statistics. you asked what the next flip is likely to be. It is an individual flip of a coin, which is always 50/50. what was flipped prior bears none on the physics of a coin flipping through the air. Yes, the odds of flipping 21 heads in a row is extremely long. However, having flipped 20 heads in a row, flipping another head is a 50/50 proposition.
Perhaps you should ckeck your facts before insulting those more intelligent than you
Ok, the odds of flipping heads again on the 9th try is equal to:
the probability that you picked a regular coin and flip 9 heads in a row,
the probability that you picked the two headed coin and flip 9 heads in a row,
this is because these are the only two ways to achieve nine straight heads according to the parameters of the ridle.
probability of picking normal coin = .9
probability of flipping 9 heads in a row with normal coin = .00195
probability of picking normal coin and then flipping nine straight heads = .9 *.00195 = .00176
probability of picking two headed coin = .1
probability of flipping nine heads on two headed coin = 1
probability of picking two headed coin and then flipping 9 straight heads = .1*1 = .1
so, overall probability that the next flip is heads = .100176
as for the bonus, each flip is independent and odds are 50/50