Maybe the coin is biased?

When you read about the Gambler's Fallacy, examples often say something like "Even if a fair coin comes up heads 10x in a row, the probability of heads on the next toss is 1/2." And to drive this home they say "In fact, even if it comes up 100x or 1000x, the odds on the next toss are 1/2."

My question is, how many heads tosses would you have to see in a row before you start thinking you're not actually looking at a fair coin? I understand that you should expect Long streaks in a lifetime of tossing coins, but is there some number that would raise your eyebrows, mathematically? Or can you look at 1000 heads in a row and still say "well, that is within expectations." How long would you put money on such a coin? Say someone was paying 2:1, a game you should play forever if fair. How many times would you have to lose before quitting (assuming that the stakes are low enough that you can afford to play as long as you want to)?

Quote: AndyGB

When you read about the Gambler's Fallacy, examples often say something like "Even if a fair coin comes up heads 10x in a row, the probability of heads on the next toss is 1/2." And to drive this home they say "In fact, even if it comes up 100x or 1000x, the odds on the next toss are 1/2."

My question is, how many heads tosses would you have to see in a row before you start thinking you're not actually looking at a fair coin? I understand that you should expect Long streaks in a lifetime of tossing coins, but is there some number that would raise your eyebrows, mathematically? Or can you look at 1000 heads in a row and still say "well, that is within expectations." How long would you put money on such a coin? Say someone was paying 2:1, a game you should play forever if fair. How many times would you have to lose before quitting (assuming that the stakes are low enough that you can afford to play as long as you want to)?

your treading in dangerous waters.....AndyGB

Quote: AndyGB

When you read about the Gambler's Fallacy, examples often say something like "Even if a fair coin comes up heads 10x in a row, the probability of heads on the next toss is 1/2." And to drive this home they say "In fact, even if it comes up 100x or 1000x, the odds on the next toss are 1/2."

My question is, how many heads tosses would you have to see in a row before you start thinking you're not actually looking at a fair coin? I understand that you should expect Long streaks in a lifetime of tossing coins, but is there some number that would raise your eyebrows, mathematically? Or can you look at 1000 heads in a row and still say "well, that is within expectations." How long would you put money on such a coin? Say someone was paying 2:1, a game you should play forever if fair. How many times would you have to lose before quitting (assuming that the stakes are low enough that you can afford to play as long as you want to)?

It varies. Have you personally examined the 'coin'? Does some entity 'guarantee' the fairness? How reputable is the guarantor? What risk does some entity take in using an unfair coin? How large is the gain that the entity will realize by using the unfair coin? If you are aware of the unfairness, can you take advantage of it? There are many other similar questions that need to be answered before your question can be answered.

In a coin flip scenario I'd always bet the streak. At worst it is 50/50 but you might get benefit of bias or cheating.

It's an established fact that the obverse or heads-side of all US coins is heavier. This is the result of the "portrait" on all US Coins.

So US coins are not "fair" for flip purposes.

And yes, Ive made numerous attempts to find out from the company that manufactures the Super Bowl coins if they have guaranteed it is a fair coin and never got a response from them.

Quote: AlanMendelson

It's an established fact that the obverse or heads-side of all US coins is heavier. This is the result of the "portrait" on all US Coins.

So US coins are not "fair" for flip purposes.

And yes, Ive made numerous attempts to find out from the company that manufactures the Super Bowl coins if they have guaranteed it is a fair coin and never got a response from them.

LOL on SB coin requests

Quote: AlanMendelson

It's an established fact that the obverse or heads-side of all US coins is heavier. This is the result of the "portrait" on all US Coins.

So US coins are not "fair" for flip purposes.

And yes, Ive made numerous attempts to find out from the company that manufactures the Super Bowl coins if they have guaranteed it is a fair coin and never got a response from them.

I would argue that it doesn't really matter if a coin is fair or not as long as the party "calling it" is ignorant of the bias. Imagine I had a double headed coin I was flipping, for instance, making it impossible for the coin to land on tails. The person calling it is still 50/50 to guess right, so nothing has changed long term

Quote: AndyGB

When you read about the Gambler's Fallacy, examples often say something like
"Even if a fair coin comes up heads 10x in a row, the probability of heads on the next toss is 1/2."
And to drive this home they say "In fact, even if it comes up 100x or 1000x, the odds on the next toss are 1/2."

My question is, how many heads tosses would you have to see in a row before you start thinking you're not actually looking at a fair coin?

How does one know with 100% certainty that a coin IS fair? Test the balance of it?
I do not know.
So I do not like a coin toss.

How about a bag with 10 identical marbles in it, or poker chips if you hate marbles,
except 5 are red marbles and 5 are blue marbles.
After shaking the bag and reaching in, not looking, remove one.
Note the color and put that marble (poker chip) back into the bag and shake and repeat.

Now would you say after picking 20 red marbles in a row the red marbles somehow are biased??
Did they grow in size or weight??
I think not. (Nice trick if they could)
And the chance of picking another red marble is somehow different from 1 in 2? (5 out of 10)

I think about it...

Nice question.
What are you leading to?
How to detect a biased coin?
Good Luck

Quote: 7craps

How does one know with 100% certainty that a coin IS fair? Test the balance of it?

Now would you say after picking 20 red marbles in a row the red marbles somehow are biased??
Did they grow in size or weight??

What are you leading to?
How to detect a biased coin?

That's what I'm thinking about, how do you know the setup is fair, or more importantly, when do you start questioning if it is fair, knowing only the nominal setup and some number of results. In the marbles example, how many red marbles showing up in a row would make you think that the real ratio was not the ratio that the person running the game told you it was, or that they weren't selecting fairly (however that would be done)?

Let's say a coin is biased and it should hit heads 51% of the time.

If you were to flip it and obtained a streak of ten consecutive heads, the chance of the next toss yielding heads would still be 51%.

Quote: AndyGB

My question is, how many heads tosses would you have to see in a row before you start thinking you're not actually looking at a fair coin?

This is a valid question, and the answer is: it depends how important a possible bias is.
In science, you usually speak about a "discovery" when you investigate something and observe a difference of 6 standard deviations from the expectation.

Applied to your coin toss example, where the standard deviation of N tosses is sqrt(N)/2: you would speak about a scientific discovery if your first 36 tosses come up all heads. Your discovery would be, that your coin tossed is not consistent with an unbiased coin and indepenedent tosses.

(double post)

Quote: treetopbuddy

your treading in dangerous waters.....AndyGB

Why is that?

Actually this is THE big fracture between classical statistics and Bayesian statistics.

State the question in reverse: how many days do you need to see the sun rise before you're "sure" the sun will rise the next day?
Now, less silly: you're in another country. How many cars do you need to see with a plate missing the letter W before concluding that they don't use W in that country's plate numbering?

In classical statistics, you need to define a confidence level. That is, up to what (small) probability do you accept that it is just a coincidence?
(But in real life, you have to add an element of psychology: what faith do you have in the person providing the coin?)
Example: a student of mine arrives late every morning. Once it's the bus, the next it's a phone call, the next a suicide in the metro, etc. Which day will I begin to judge he is pulling my leg? The answer depends on what I know of him in a general context, how much I trust him/her.

In Bayesian statistics, you start with a neutral probability distribution and adapt it with each observed result. The more heads you see, the more you adapt your (posterior Bayesian or subjective) probability towards heads. So, in that view, NO coin is ever equilibrated. So the question becomes: at what level of bias do I judge I'm being cheated. That is somehow equivalent to fixing a confidence level.

Quote: MangoJ

Applied to your coin toss example, where the standard deviation of N tosses is sqrt(N)/2: you would speak about a scientific discovery if your first 36 tosses come up all heads. Your discovery would be, that your coin tossed is not consistent with an unbiased coin and indepenedent tosses.

Thanks MangoJ, this is exactly what I was looking for, in terms of letting math have a say over intuition. I feel like my personal intuition threshold, and maybe that of a lot of other people too, would be triggered a lot sooner, like 15 in a row.

Quote: kubikulann

Actually this is THE big fracture between classical statistics and Bayesian statistics.

State the question in reverse: how many days do you need to see the sun rise before you're "sure" the sun will rise the next day?
Now, less silly: you're in another country. How many cars do you need to see with a plate missing the letter W before concluding that they don't use W in that country's plate numbering?

In classical statistics, you need to define a confidence level. That is, up to what (small) probability do you accept that it is just a coincidence?
(But in real life, you have to add an element of psychology: what faith do you have in the person providing the coin?)
Example: a student of mine arrives late every morning. Once it's the bus, the next it's a phone call, the next a suicide in the metro, etc. Which day will I begin to judge he is pulling my leg? The answer depends on what I know of him in a general context, how much I trust him/her.

In Bayesian statistics, you start with a neutral probability distribution and adapt it with each observed result. The more heads you see, the more you adapt your (posterior Bayesian or subjective) probability towards heads. So, in that view, NO coin is ever equilibrated. So the question becomes: at what level of bias do I judge I'm being cheated. That is somehow equivalent to fixing a confidence level.

Thanks for the info on this, I'm not at all familiar with Bayesian statistics, I will have to read up more!

Hard to find perfection in this world.
Anything built by humans should not be perfect.
For the initial question: you must follow the six-sigma rule to estimate the randomnness or the bias of the coin.
In 100 tosses of a fair coin you need 70 hit to 30 misses to get +4 standard deviations.
Supose you test the coin once and get +65(3st dev) and -35, another test breaking 2 to 3 standard deviations doblechecks that this coin has some probles.ç
The other realm is to determina how biased it is.
Bayessian rules might work here.
We need a math guy to give the accurate numbers.

Quote: ybot

For the initial question: you must follow the six-sigma rule.

Why "must"? There is no law here. Just a cultural habit. If I want, I may choose 5 sigma, or 7 sigma.

Wouldn't you have to compare relative probabilities of random events happening?

Chance of getting blackjack in single deck game:
1 in 21 = approx. 4 heads in a row

Chance of pregnancy having identical twins:
1 in 333 = approx. 8 heads in a row

Chance of being hit by lightning in a lifetime:
1 in 10,000 = approx. 13 heads in a row

Chance of flopping a royal flush:
1 in 649,739 = approx. 19 heads in a row

Chance of winning the Mega Millions lottery:
1 in 175,711,535 = approx. 27 heads in a row

Chance of winning 56 of 93 hands in a Texas Hold’em tournament with 8 equally skilled players:
1 in 1.88 x 10^44 = approx. 147 heads in a row = 6 standard deviations