Always 50/50 chance?

[drkitten]

A number of people have stressed independence of the coin tosses. I'm not sure why. Can someone explain?

Because if the coin tosses are not independent -- of course, if they're not independent, they're not really coin tosses -- then one can improve on the straight-up probability calculations by doing context-dependent guessing.

As a simple if extreme example, if I know that the next event will always be the opposite of this one, the optimal strategy is to alternate guessing heads and tail. Not, I hasten to point out, to choose randomly between the two, but to alternate. No randomness is involved.

The expectation of the sum of a bunch of random variables is equal to the sum of the expectations of the random variables, whether the random variables are independent or not.

But if the random variables are not independent, then the expectations are not independent either, and one can generally do better than "chance" with a strategy that takes advantage of the non-independence of the expectations.

For example, we've been discussing both coin flips and sex of children as though they were random. I believe that is true for coin flips; I do not believe that to be true for children. Specifically, I think there is some evidence that not all men produce equal numbers of X- and Y- chromosome sperm, which in turn implies a positive correlation between the sex of siblings. I pawed through some PubMed cites and couldn't find anything to confirm this -- so either take my word for it, or else treat this as a hypothetical.

But if this is the case, I could actually predict the sex of a child with better than 50% accuracy if that child has an older sibling. Simply predict that the younger child will have the same sex as the elder.

And, of course, this reduces to an absurdity if you walk into my office and ask me to predict the sex of your identical twin sibling. Despite the fact that twins are, in general, distributed uniformly across the sexes, I can tell you what your identical twin's sex is with little uncertainty.

 

[drkitten]

You need independence for these arguments. Take the case where I have a bag with 5 red balls and 5 blue balls. I now draw each ball out of the bag one at a time.

Person A guesses: RBRBRBRBRB
Person B guesses: RRRRRRRRRR
Person C guesses: BBBRBBBRBB

[...]

Because the variances are different, the "optimal" guessing strategies can possibly be different.

But also, the "optimal" guessing strategy cannot be pre-specified. In particular, I should never miss on the last ball drawn -- but my guess will be conditioned on the previous nine.

 

[JonnyFive]

Eek! Did I drop an "odds" comment when I should have said "prob"??? Well spank me sideways with a soggy tuna. Do I have to turn in my "Stats Geek" membership card now or at the door?

I don't remember. I saw a couple people do it, and I figured I'd post for Hammer's edification, if nothing else.

It's probably also worth noting that for very low probability events, the odds and the probability are often functionally the same, for any practical purposes.

As Home Simpson once said: "Correction - a million to fifty!"

 

[rtalman]

Well, if you know that p(Girl) is 75%, then shouldn't you change the random guess to also be 75% Girl? Then at least you're slightly closer to my optimal solution of always guessing "Girl"

Yes, I probably should have changed my random to bias to 75% girl.

But back to the OP, if you take a large enough sample where you can make a +/- guess on an outcome, then the odds are 50/50 for just about anything empirically measurable. Not the odds of the outcome, the odds of you randomly guessing correctly 50% of the time, provided you keep your guesses random. So if you take a population of 1,000 people and try to guess girl/boy by only looking at each person's social security number, you can guess randomly and come out 50/50, even if the population is biased 99.9% toward a specific gender.

In a poker analogy, you will get dealt pocket aces about one out of every 221 hands. You are sitting at a casino for 24 hours straight, if the dealer button is at an odd seat, tell yourself you are going to get pocket rockets. If the button is at an even seat, tell yourself you are not going to get pocket rockets. You can predict with a 50% accuracy rate something that only happens .45% of the time.

Wow, I am going to get my application in for the MDC before it is too late!

 

[Baron Samedi]

I don't remember. I saw a couple people do it, and I figured I'd post for Hammer's edification, if nothing else.

It's probably also worth noting that for very low probability events, the odds and the probability are often functionally the same, for any practical purposes.

As Home Simpson once said: "Correction - a million to fifty!"

Heh. I used to go around and try to correct people on that. I stopped after having to tell people way too often that 2:1 odds does not mean a 50% chance of winning. Again, great people to play poker with.

 

[NobbyNobbs]

I have a math question.

Is anything not a 50/50 chance of being true, right, or correct?

As an atheist I believe that there is no god but there may be. There is a 50/50 chance that god exists though. As with anything else. I will either wake up tomorrow or I wont. See how it goes

My thinking may be over simple so I ask for some help in working through this.

Thanks

There's a 50% chance you're right about this.

 

[RecoveringYuppy]

Because if the coin tosses are not independent -- of course, if they're not independent, they're not really coin tosses -- then one can improve on the straight-up probability calculations by doing context-dependent guessing.

hmmmm. 69doge might be trying to make a different point.

Statistical independence can be satisfied in systems where context-dependent guessing is possible. The requirement for SI is merely that context-dependent guessing not be happening, not whether it's possible. It's not strictly necessarily that the coin tosses be independent of earlier coin tosses. The requirement is that that the "guesser" be unaware or incapable of taking advantage of that dependence*. IOW the predictor and the system being predicted must be independent of each other, but both the predictor and the system being predicted may be dependent on their own internal state.

So, if that's what 69dodge has mind, the answer to his question is because we all appear to be discussing a case where one system is capable of remembering and reacting to the state of the other system.

Hope that makes sense.

Edited to add footnote for * above: Of course it goes without saying that the coin need also be unaware or incapable of reacting to the guess being made.

 

[Baron Samedi]

Yes, I probably should have changed my random to bias to 75% girl.

But back to the OP, if you take a large enough sample where you can make a +/- guess on an outcome, then the odds are 50/50 for just about anything empirically measurable. Not the odds of the outcome, the odds of you randomly guessing correctly 50% of the time, provided you keep your guesses random. So if you take a population of 1,000 people and try to guess girl/boy by only looking at each person's social security number, you can guess randomly and come out 50/50, even if the population is biased 99.9% toward a specific gender.

In a poker analogy, you will get dealt pocket aces about one out of every 221 hands. You are sitting at a casino for 24 hours straight, if the dealer button is at an odd seat, tell yourself you are going to get pocket rockets. If the button is at an even seat, tell yourself you are not going to get pocket rockets. You can predict with a 50% accuracy rate something that only happens .45% of the time.

Wow, I am going to get my application in for the MDC before it is too late!

Ah, that's the super fun part!

If the actual outcome is 50/50, your accuracy will be 50% no matter what.
If your guessing strategy is 50/50, your accuracy will also be 50% no matter what the base is.

That 50% is such a magical number.
*weeps in joy*

 

[69dodge]

Because if the coin tosses are not independent -- of course, if they're not independent, they're not really coin tosses -- then one can improve on the straight-up probability calculations by doing context-dependent guessing.

As a simple if extreme example, if I know that the next event will always be the opposite of this one, the optimal strategy is to alternate guessing heads and tail.

Ok, so you know that the "coin tosses" will strictly alternate heads and tails.

If you've already seen one, you can guess perfectly from now on. But, also, if you've already seen one, the probability is no longer 50% that the next toss (or any given subsequent toss) will be a head, because you know what it will be.

If you haven't seen one yet, each of the future tosses does still have a 50% probability of being a head. But, also, if you haven't seen one yet, you don't know whether to guess HTHT... or THTH... .

See what I mean?

 

[Dave1001]

Great sig quote by the way, RY

"Make a fire for a man and you keep him warm for a day. Set him on fire and you keep him warm for the rest of his life."

 

[The Atheist]

I believe that is true for coin flips; I do not believe that to be true for children. Specifically, I think there is some evidence that not all men produce equal numbers of X- and Y- chromosome sperm, which in turn implies a positive correlation between the sex of siblings. I pawed through some PubMed cites and couldn't find anything to confirm this -- so either take my word for it, or else treat this as a hypothetical.

I'm Don't we know with a reasonable degree of assurance that babies aren't 50:50 boy/girl anyway? The results over quite a long period of time suggest that it's closer to 51:49, M:F.

Another point to consider is that X & Y sperm have different characteristics and possibly speeds. Add to that ovulation and sexuality and I think "random" doesn't make first base.

This woman claims to have "proven" a 94.7% success rate in choosing the sex of your baby! (sounds hysterically high)

 

[Mangafranga]

GGGB vs GGGG and BBBB

Clearly the correct result has been covered, but I didn't see this solution offered. This solution is offered because I think it is quite intuitive.

If GGGB is worse than GGGG and BBBB, then either in the GGGG set the first 3 G's must perform worse than the 4th G, or the in the BBBB set the 4th B must perform worse than the first 3 Bs (or both.)

Hopefully that is correct (lest I encourage 2 more pages )

Got the better/worse thing the wrong way round
And then didn't change all instances of it, no wonder this gets so confusing

 

[Gilmar]

B. Kliban (I think) did a cartoon showing possible outcomes of a coin toss:
It lands Heads-up
It lands Tails-up
It lands on its edge
It is nabbed in mid-air by a passing bird and does not come back down.
(Or was it Shel Silverstein?)

 

[Dave1001]

GGGB vs GGGG and BBBB

Clearly the correct result has been covered, but I didn't see this solution offered. This solution is offered because I think it is quite intuitive.

If GGGB is worse than GGGG and BBBB, then either in the GGGG set the first 3 G's must perform worse than the 4th G, or the in the BBBB set the 4th B must perform worse than the first 3 Bs (or both.)

Hopefully that is correct (lest I encourage 2 more pages )

Got the better/worse thing the wrong way round
And then didn't change all instances of it, no wonder this gets so confusing

You're right. Great, intuitive explanation. I wonder if there's a specific term for the sort of cognitive bias that a few of us experienced to think GGGB would be wrong more often than BBBB or than a random guessing pattern with 50% G's and 50% B's -probably 2 different cognitive biases.

 

[Baron Samedi]

Ok, so you know that the "coin tosses" will strictly alternate heads and tails.

If you've already seen one, you can guess perfectly from now on. But, also, if you've already seen one, the probability is no longer 50% that the next toss (or any given subsequent toss) will be a head, because you know what it will be.

If you haven't seen one yet, each of the future tosses does still have a 50% probability of being a head. But, also, if you haven't seen one yet, you don't know whether to guess HTHT... or THTH... .

See what I mean?

But let's say that as of this moment, you don't know if the last toss was H or T. Therefore, the next toss still has a 50% chance of being Heads, and a 50% chance of being Tails. And you must call the next 10 tosses NOW. So you're either going to pick HTHTHTHTHT or THTHTHTHTH. Your accuracy will therefore be 0 or 10 out of 10, nothing inbetween. Play it enough times, sure, your AVERAGE score is 5/10, and each toss is indeed 50% heads, 50% tails, but this certainly does not behave anywhere near the same as independent coin tosses.

Stressing independence just makes the calculations sooooooooooooooo much easier, and then you don't have to worry about all these wacky and crazy cases. If you have dependence, you can cheat and manipulate and get any old results you see fit. So to go back to the original point of a psychic guessing coin tosses with better than 50% accuracy... if the tosses are independent, there's possibly something here. If the tosses are dependent, then this so-called psychic is probably cheating by knowing the dependencies. Getting 10 out of 10 on a random independent coin tosses has a probability of 0.1%. Getting 10 out of 10 on the above example is 50%. Randi's money would have been gone ages ago if we allowed this.

 

[Jekyll]

But let's say that as of this moment, you don't know if the last toss was H or T. Therefore, the next toss still has a 50% chance of being Heads, and a 50% chance of being Tails. And you must call the next 10 tosses NOW. So you're either going to pick HTHTHTHTHT or THTHTHTHTH. Your accuracy will therefore be 0 or 10 out of 10, nothing inbetween. Play it enough times, sure, your AVERAGE score is 5/10, and each toss is indeed 50% heads, 50% tails, but this certainly does not behave anywhere near the same as independent coin tosses.

And of course, saying "Play it enough times" is just another way to crowbar the independence back in to the calculations.

 

[strathmeyer]

This thread was awesome until after post #18. Then it was ruined by the slow witted. Then it was ruined by the nerds.

 

[The Atheist]

Then closed at #137 by a pompous twat?

 

[Jekyll]

138.

 

[69dodge]

But let's say that as of this moment, you don't know if the last toss was H or T. Therefore, the next toss still has a 50% chance of being Heads, and a 50% chance of being Tails. And you must call the next 10 tosses NOW. So you're either going to pick HTHTHTHTHT or THTHTHTHTH. Your accuracy will therefore be 0 or 10 out of 10, nothing inbetween. Play it enough times, sure, your AVERAGE score is 5/10, and each toss is indeed 50% heads, 50% tails, but this certainly does not behave anywhere near the same as independent coin tosses.

Right. It doesn't behave the same in all ways. It just has the same expectation.

Stressing independence just makes the calculations sooooooooooooooo much easier, and then you don't have to worry about all these wacky and crazy cases.

Some calculations are different. Calculation of the expectation is the same. Or, at least, it is the same if you do it a certain way.

I'd say, it's easier in this case not to rely on independence. Just add the individual expectations to find the expectation of the sum.

If you have dependence, you can cheat and manipulate and get any old results you see fit. So to go back to the original point of a psychic guessing coin tosses with better than 50% accuracy... if the tosses are independent, there's possibly something here. If the tosses are dependent, then this so-called psychic is probably cheating by knowing the dependencies. Getting 10 out of 10 on a random independent coin tosses has a probability of 0.1%. Getting 10 out of 10 on the above example is 50%. Randi's money would have been gone ages ago if we allowed this.

Yes, but...

I'm not sure I'd call that cheating or manipulating. It's still just pure guessing. In this case, there's a 50% probability that guessing will result in 10 correct and there's a 50% probability that guessing will result in 0 correct.

I don't think this is the sort of thing Dave1001 had in mind when he suggested that guessing HHHT... is "worse" than guessing some other sequence. He wasn't very specific, but I figure that if HHHT... had a high probability of being very bad but also had an equally high probability of being very good, it shouldn't be considered worse overall. Nor better, of course. Just different.