Probability Question

This isn't really anything intellectually stimulating, but rather a question that I need to know the answer to...

My question is: If you flip a coin 3 times, and all 3 times it lands on heads, does it become more likely that it'll land on tails if you flip the coin again?

I thought that the probability still remains the same (50% chance of heads or tails) but my friend doesn't think so, he thinks it'll be more likely to land on tails when the coin is flipped again.

Can anyone offer some clarification to this?

Originally posted by Dance Master MC
This isn't really anything intellectually stimulating, but rather a question that I need to know the answer to...

My question is: If you flip a coin 3 times, and all 3 times it lands on heads, does it become more likely that it'll land on tails if you flip the coin again?

I thought that the probability still remains the same (50% chance of head ...[text shortened]... and on tails when the coin is flipped again.

Can anyone offer some clarification to this?

The probability stays at 50/50, of course, assuming you have a fair coin. The coin has no "memory": past flips cannot affect future flips.

Originally posted by GregM
The probability stays at 50/50, of course, assuming you have a fair coin. The coin has no "memory": past flips cannot affect future flips.

Thank you for the reassurance, I knew I wasn't stupid....

Everytime you flip the coin there is a 50 50 chance of head / tail

The stipulations says that the coin is fair. Suppose you don't know that.

So you have a coin (or a irregular stone or something) that you don't know anything about its fairness. You flip it thrice with the same result. Is this a proof that the coin is not fair? Can you tell anything about its fairness after this trhe flips?

What about the fourth time - do you know anything about the fourth outcome, now that you don't know anything about the fairness of the coin?

Originally posted by FabianFnas
The stipulations says that the coin is fair. Suppose you don't know that.

So you have a coin (or a irregular stone or something) that you don't know anything about its fairness. You flip it thrice with the same result. Is this a proof that the coin is not fair? Can you tell anything about its fairness after this trhe flips?

What about the fourth tim ...[text shortened]... ng about the fourth outcome, now that you don't know anything about the fairness of the coin?

If we keep throwing the coin, we'll eventually be able to use a binomial test, for example.

Don't you hate people who tell you untrue things like that? And actually believe themselves! I think the best way to beat his 'theory' is that if he was right, casinos would be losing money!

Interesting question, Fabian. If it flipped three times in a row heads, I would still be very sure the coin was fair. However, if the coin was flipped 100 times heads in a row, i would be willing to bet that the coin was double-headed. Each person has a different 'threshold', the time when they start believe the coin is biased. I don't think there is a mathematical answer,

Originally posted by Dejection
I think the best way to beat his 'theory' is that if he was right, casinos would be losing money!

Casinos earn money without great risk, because the law of large numbers guarantees that, as more and more games are played, you converge towards the expected value. If there was such a mechanism that increased this convergence speed towards the expected value, then casinos would actually decrease their risk and benefit from it (as risk is undesirable).

You have 2 coins; one is a normal, fair coin, the other is double-headed.

You randomly pick one of the coins and pass to a friend who flips it three times noting the result.

If the result is HHH what is the probability the coin is fair?

Originally posted by wolfgang59
You have 2 coins; one is a normal, fair coin, the other is double-headed.

You randomly pick one of the coins and pass to a friend who flips it three times noting the result.

If the result is HHH what is the probability the coin is fair?

1/9?

Yep .. thats what I got.

Originally posted by wolfgang59
Yep .. thats what I got.

Took me two edits to get it right, though.

Originally posted by Dejection
Each person has a different 'threshold', the time when they start believe the coin is biased. I don't think there is a mathematical answer,

Naturally there is a mathematical answer, bayesian statistics.

The answer of how likely a coin is likely to come up heads again based on Baynesian statistics also requires an approximation of the odds the coin is fair.

We've seen two existing cases and their answers.

If the odds for a fair coin is 100%, the odds for another heads will be 50%.

If half the coins are double-headed by our best knowledge, then based on the results, the chances we grabbed an unfair coin if 8/9, which makes the odds for a fourth head at 17 out of 18.

If 99% coins are fair, then we have a 99/107 ( 92.5% ) chance the coin is fair, and a result of 115/214 ( 53.7% ) chance the next toss is heads.

Originally posted by Dance Master MC
Thank you for the reassurance, I knew I wasn't stupid....

It would be a different question altogether if you asked what are the odds of flipping 4 heads in a row?

The odds are much lower than 50% of flipping 4 heads in a row.

This may be what your friend is getting at.


By telling us 3 heads have ALREADY been flipped, you are simplifying the question to a simple coin toss. :>