Solving the Coin Flips Problem - Probability of Tails After 9 Heads

[joebloggs]

Hi, If I flip a fair coin 9 times and each time it comes up heads what is the probability that on the tenth time I flip the coin it will come up tails?

At first I thought the probability that the coin would come up tails would be 50% as each coin flip is an independent event. But I was thinking surely that the probability of flipping a tail after 9 heads should be greater than 50% right?

This is isn't a homework question (I'm too old to go to school!) it's just something that has been bugging me!

 

[CRGreathouse]

A fair coin is one that comes up heads 50% and tails 50%, so the answer is 50%.

If you saw a coin come up heads 9 times in a row, you might question whether it was a fair coin. If it's a weighted coin, it's probably weighted to come up heads more often, in which case the chance of heads is more than 50%. I wouldn't ever guess the chance of tails as more than 50% in that situation.

 

[Mentallic]

While I understand probabilities quite well (at a senior high school level), I still allow myself to become consumed with these types of contradictions. While reading through your question, and seeing where you were going with it before actually getting to the part:

But I was thinking surely that the probability of flipping a tail after 9 heads should be greater than 50% right?

I rather thought you were going to say the opposite. If 2 events are supposedly of equal probability (as this coin should be) but one keeps happening while the other never does. Wouldn't it also be safe to assume that the next event will be the one that has been happening consistently (heads) rather than the one that has yet to occur (tails).
Some might start to become skeptical at the idea that the events might not be of equal probabilities (while theoretically they are still equal).

 

[lurflurf]

While I understand probabilities quite well (at a senior high school level), I still allow myself to become consumed with these types of contradictions. While reading through your question, and seeing where you were going with it before actually getting to the part:

I rather thought you were going to say the opposite. If 2 events are supposedly of equal probability (as this coin should be) but one keeps happening while the other never does. Wouldn't it also be safe to assume that the next event will be the one that has been happening consistently (heads) rather than the one that has yet to occur (tails).
Some might start to become skeptical at the idea that the events might not be of equal probabilities (while theoretically they are still equal).

It depends on ones assumptions.
The OP stated that the coin was fair.
No amount of trials can determine a probability with any certainty.
The only statemets that can be made are those linking probability to likely outcomes.
Say we flip a coin of unknown fairness one million times and get heads each time.
We might say things like
If that was a fair coin, that outcome was very unusual.
If that was a very stongly heads favoring coin the result was usual.
If that was a very stongly tails favoring coin the result was exceptionally unusual.

We can use the outcome to eastamate the fairness, but only in a roundabout way, and only with additional assumptions.