Law of large numbers regression to mean

[Dg1]

I’ve been away from math for a long time so I hope I make sense and this is probably an easy question for the intelligent people on here.
If you flip a coin and the odds are fifty percent heads or tails for one flip. How do calculate when the coin is flipped say 10 times it’s all heads. It’s fifty percent still for the next single flip but how is the law large numbers and regression to mean calculated in? Thanks

 

[khansaheb]

How do calculate when the coin is flipped say 10 times it’s all heads

Calculate what?

 

[Dr.Peterson]

How do calculate when the coin is flipped say 10 times it’s all heads.

Do you mean, how do you calculate the probability that all 10 flips will be heads?

The first will be heads 1/2 of the time; the first two will both be heads 1/2 of that 1/2, or 1/4 of the time; and so on. So all 10 will be heads (1/2)^10 = 1/1024 of the time. That's the probability.

This is the product rule for probabilities: P(A and B) = P(A) * P(B), if events A and B are independent (as coin flips are).

But where does regression to the mean come into your question? The law of large numbers just says that if you repeat an experiment many times, the experimental probability will be close to the theoretical probability -- so the probability we calculate only approximates what we'd actually observe.

 

[Dg1]

Do you mean, how do you calculate the probability that all 10 flips will be heads?

The first will be heads 1/2 of the time; the first two will both be heads 1/2 of that 1/2, or 1/4 of the time; and so on. So all 10 will be heads (1/2)^10 = 1/1024 of the time. That's the probability.

This is the product rule for probabilities: P(A and B) = P(A) * P(B), if events A and B are independent (as coin flips are).

But where does regression to the mean come into your question? The law of large numbers just says that if you repeat an experiment many times, the experimental probability will be close to the theoretical probability -- so the probability we calculate only approximates what we'd actually observe.

Thank you that’s is exactly why I was looking for. Trying to wrap my head around the concept of regression to mean was throwing me off. Reminds me of schroedinger’s cat.so if there is regression Is there an average amount that is considered “around many” that things begin to average out to a point that it slows down trying to get back to 1/2 and stays in a percentage range then? This is very interesting. I’ve never played with math just for fun. Thank you again.

 

[Dg1]

Calculate what?

Sorry. Just meant what would the probability be of it being heads the next flip and how to calculate that.

 

[Dr.Peterson]

Sorry. Just meant what would the probability be of it being heads the next flip and how to calculate that.

It's important to realize that this is not the same question I answered:

Do you mean, how do you calculate the probability that all 10 flips will be heads?

Thank you that’s is exactly why I was looking for.

The probability that the next flip will be heads is always 1/2 (regardless of how many have been heads); the probability that all ten are heads is very small. The latter was your question:

How do calculate when the coin is flipped say 10 times it’s all heads.

Is that clear?

 

[Dg1]

Yeah. That clears it up. Thank you. I was just looking for (1/2)^10= 1/1024. Just worded things wrong.

 

[Dg1]

It's important to realize that this is not the same question I answered:



The probability that the next flip will be heads is always 1/2 (regardless of how many have been heads); the probability that all ten are heads is very small. The latter was your question:


Is that clear?

In a case like this where the law of small numbers or gamblers fallacy makes the probability of 10 in row almost useless in helping predict future outcomes. What could the probability of a small number calculation like this be used for in real world applications?

 

[BigBeachBanana]

In a case like this where the law of small numbers or gamblers fallacy makes the probability of 10 in row almost useless in helping predict future outcomes. What could the probability of a small number calculation like this be used for in real world applications?

1) I don't see how the law of small numbers applies here.
2) Small probability doesn't mean "useless", why would you say it is useless? It's the opposite.
3) In real-world applications, a small probability tells you you shouldn't buy the PowerBall where the odd is 1 in 292,201,338. It's better to put your money in savings and earn risk-free money.

 

[Dg1]

1) I don't see how the law of small numbers applies here.
2) Small probability doesn't mean "useless", why would you say it is useless? It's the opposite.
3) In real-world applications, a small probability tells you you shouldn't buy the PowerBall where the odd is 1 in 292,201,338. It's better to put your money in savings and earn risk-free mon

 

[Dg1]

A large lottery is a great example of application thank you.
I didn’t mean “totally useless” I meant “almost”. I was just looking at the way casinos rely on the gamblers fallacy and how past events in a series doesn’t affect future events. In my small brain If the odds of flipping 10 heads in a row are 1/1024 and you’ve already flipped nine you should have a better chance of flipping tails, but you don’t, the 9 already happened so it’s still 1/2. I understand it. I just doesn’t seem logical, but then again “seem” isn’t always trustworthy.

 

[Dr.Peterson]

In a case like this where the law of small numbers or gamblers fallacy makes the probability of 10 in row almost useless in helping predict future outcomes. What could the probability of a small number calculation like this be used for in real world applications?

It appears that "law of small numbers" is used in a couple different ways; how are you using it?

I didn’t mean “totally useless” I meant “almost”. I was just looking at the way casinos rely on the gamblers fallacy and how past events in a series doesn’t affect future events. In my small brain If the odds of flipping 10 heads in a row are 1/1024 and you’ve already flipped nine you should have a better chance of flipping tails, but you don’t, the 9 already happened so it’s still 1/2. I understand it. I just doesn’t seem logical, but then again “seem” isn’t always trustworthy.

My own perspective on the gambler's fallacy is that, if I see someone tossing a coin ten times and every flip is heads, I would want to examine the coin and see if it has a tails side -- that is, I'd check my assumptions. That amounts to statistical hypothesis testing.

And regression to the mean doesn't mean that past events affect future events; it's just that in the long run, everything will tend to balance out just by continuing as usual. If the next million flips are 50:50, then the million-and-ten flips will be closer to 50:50 than the first ten were.

But if it's a biased coin, then all bets are off.

 

[Agent Smith]

What's the law of small numbers?

I sometimes feel math gets tooooo... theoretical. Has anyone actually run a test on the gambler's fallacy? One that could be statistically analyzed to check if what's in books matches what's ... the truth is ... out there?

I'll try a simulation and get back to you guys ...

The gambler's gambit post-losing-streak doesn't work because no matter how "big" the streak is, the universe has [imath]\infty[/imath] to even the score. Too, from the little that I know of probability, each streak is an "entity" unto itself. So 9 heads in a row is ONE event with a probabilty of [imath]\left(\frac{1}{2}\right)^9[/imath], just as any other combination of heads & tails, which includes 9 tails in a row and 8 heads and 1 tail, is

The number [imath]\frac{1}{2}[/imath] is, how shall I put it?, weird. Any other number/fraction and this sort of "equilibrium" is not seen.

The event 9-heads-in-a-row is improbable, but so is every other combination of heads & tails for 9 flips.

 

[Agent Smith]

What's the law of small numbers?

I sometimes feel math gets tooooo... theoretical. Has anyone actually run a test on the gambler's fallacy? One that could be statistically analyzed to check if what's in books matches what's ... the truth is ... out there?

I'll try a simulation and get back to you guys ...

The gambler's gambit post-losing-streak doesn't work because no matter how "big" the streak is, the universe has [imath]\infty[/imath] to even the score. Too, from the little that I know of probability, each streak is an "entity" unto itself. So 9 heads in a row is ONE event with a probabilty of [imath]\left(\frac{1}{2}\right)^9[/imath], just as any other combination of heads & tails, which includes 9 tails in a row and 8 heads and 1 tail, is

The number [imath]\frac{1}{2}[/imath] is, how shall I put it?, weird. Any other number/fraction and this sort of "equilibrium" is not seen.

The event 9-heads-in-a-row is improbable, but so is every other combination of heads & tails for 9 flips.

Ran a sim, nope the gambler's gambit is not gonna work. It's 50/50, just as the math shows.

 

[khansaheb]

I sometimes feel math gets tooooo... theoretical.

I think the math gets too logical - and we forget the assumptions (stated and unstated) that has been made.

 

[Agent Smith]

I think the math gets too logical - and we forget the assumptions (stated and unstated) that has been made.

Too logical ... yep!

 

[Dg1]

It appears that "law of small numbers" is used in a couple different ways; how are you using it?

My own perspective on the gambler's fallacy is that, if I see someone tossing a coin ten times and every flip is heads, I would want to examine the coin and see if it has a tails side -- that is, I'd check my assumptions. That amounts to statistical hypothesis testing.

And regression to the mean doesn't mean that past events affect future events; it's just that in the long run, everything will tend to balance out just by continuing as usual. If the next million flips are 50:50, then the million-and-ten flips will be closer to 50:50 than the first ten were.

But if it's a biased coin, then all bets are off.

At 1/2 wouldn’t progression to the mean be the opposite and equal of progression from the mean? In my head I just see a graph line going up and down in a chaotic infinite fashion with no mean line that it would it want to bounce around at a 50% or 1/2 value at a percentage value lower or higher the line I’ll call it chaos line while still bouncing around would have a trend to go up or down accordingly though? I’ll keep reading and learning on this. Do you have a preferred resource to learn about this? Thank you for your help and patience in understanding. You seem to be a very learned individual.

And yes i’d want check the coin if it was heads 10 times in a row also lol . If it wasn't rigged I’m still dumb enough to bet on tails for the next flip though.

 

[Dr.Peterson]

At 1/2 wouldn’t progression to the mean be the opposite and equal of progression from the mean? In my head I just see a graph line going up and down in a chaotic infinite fashion with no mean line that it would it want to bounce around at a 50% or 1/2 value at a percentage value lower or higher the line I’ll call it chaos line while still bouncing around would have a trend to go up or down accordingly though? I’ll keep reading and learning on this. Do you have a preferred resource to learn about this? Thank you for your help and patience in understanding. You seem to be a very learned individual.

The phrase is "regression to[ward] the mean"; neither of your phrases is standard, to my knowledge, so I don't know what you're trying to say.

I recommend you read up on what the phrase means, perhaps starting with Wikipedia, where I often start (then taking links from there, or searching for terms I learn there). I don't think you have a clear idea of its meaning. (There's nothing special about 1/2 in this regard.)

All it means is that if something unusual happens, then something more usual is likely to happen next. This is obvious, but has interesting implications.

Here's a nice short article I found (by searching for images referencing the phrase, since you mentioned a graph). It's also mentioned in this article on my own site about the gambler's fallacy, which you may find helpful.

 

[Dg1]

I got my terminology messed up sorry. Makes it hard to ask a question that doesn’t make sense when the terms don’t even make sense.
Thank you for the links they greatly appreciated. I’ll do some reading and maybe figure out what I’m trying to ask.