Teaching an intelligent guy about a basic statistics concept
 

A friend of mine who has a decent understanding of statistics believes that since random results will average out in the long run, when there's a variation from the standard, it will tend to go the other way in the future.

For example, he believes that if you flip a coin 10 times and get 10 heads, you're slightly more likely to get a tails next time since 11 heads in a row is very unlikely.

This is an intelligent guy, but I can't get to him on this. I'm not a good teacher.

What approach do you reccomend?

I've tried the "you're adjudicating a mystical power to the coin", the "the past doesn't influence the future in this way", and the "new information about the way things went affects how you project the long term result, so you can't expect to go 100 heads and 100 tails on average for 200 flips after you've gotten 10 heads in a row"; but none of these seem to work.

Re: Teaching an intelligent guy about a basic statistics concept
 

Your friend is not intelligent.

Edit: So, you take a fair coin and 10 flips in a row are heads. He now can have one of three assumptions:

1) Heads is more likely on the next flip
2) Tails is more likely on the next flip
3) They are equally likely

Ask him why he doesn't think heads is more likely?

Re: Teaching an intelligent guy about a basic statistics concept
 

Offer to bet him. Say you'll repeatedly. flip a coin, and every time it comes up five in a row heads or tails, you will bet on the same result for the sixth toss, as long as he gives you 11:10.

At least if you can't convince him you'll make some money.

Re: Teaching an intelligent guy about a basic statistics concept
 

Intelligent people can believe pretty silly things (like the medical doctor I met who insisted that Martingale betting was risk free easy money). The problem is that they tend to have too much of an ego about such things to admit that they're wrong about something so simple.

Some ideas though:
1) ask him to calculate the odds of a fair coin coming up heads once
2) ask him to calculate the odds of it coming up heads twice
3) ask him to calculate the odds of it coming up heads 10 times in a row
4) ask him to calculate the odds of it coming up heads one more time after it has come up heads 10 times in a row
5) ask him to calculate the odds of it coming up heads 11 times in a row

Now see if he applies probability theory consistently as he does this.

Re: Teaching an intelligent guy about a basic statistics concept
 

Bring it down to simple physics, and ask him if the movement of your fingers or the air currents in the room are going to change depending on whether the last throw was heads or tails.

Re: Teaching an intelligent guy about a basic statistics concept
 

The law of probability doesn’t say random results will even out in the long run. The more coin flips that you have, the more likely you will APPROACH the true odds.

Example: if the first 10 flips are heads and every other flip is tails, heads, tails, etc, then after 100 flips, you will have 55/45 or heads will come up 55 percent. After 1000 flips you will have 505/495 or heads will come up 50.5 percent of the time. The more flips you have the closer the odds of heads coming up will be 50% even though, at all times, the first 10 flips were heads.

Re: Teaching an intelligent guy about a basic statistics concept
 

This seems like a really easy problem. Just ask him if he thinks anyone ever flipped that coin before the cashier at the supermarket gave it to him.

Re: Teaching an intelligent guy about a basic statistics concept
 

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Example: if the first 10 flips are heads and every other flip is tails, heads, tails, etc, then after 100 flips, you will have 55/45 or heads will come up 55 percent. After 1000 flips you will have 505/495 or heads will come up 50.5 percent of the time. The more flips you have the closer the odds of heads coming up will be 50% even though, at all times, the first 10 flips were heads.

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I'm gonna try this one and let you guys know. It seems like it will work.

Thanks for the help.

Re: Teaching an intelligent guy about a basic statistics concept
 

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A friend of mine who has a decent understanding of statistics believes that since random results will average out in the long run, when there's a variation from the standard, it will tend to go the other way in the future.

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it doesn't have to go the other way in the future...imagine it goes 100 tails in a row to start...now it's 100% tails and 0% heads...he's thinking that there should now be 100 more heads than tails over the next 'long run' set of flips to 'even it out'

that's incorrect


what if it now goes 10mil heads to 10mil tails? that 100 no longer is significant...10mil heads to 10mil+100 tails is still 50/50 (49.99975% heads)...get it to billions etc and it's even closer

so no, it doesn't have to 'even out'...

and no matter how u look at it, there is no memory, so it's always 50/50 in any individual coinflip

Re: Teaching an intelligent guy about a basic statistics concept
 

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A friend of mine who has a decent understanding of statistics believes that since random results will average out in the long run, when there's a variation from the standard, it will tend to go the other way in the future.

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it doesn't have to go the other way in the future...imagine it goes 100 tails in a row to start...now it's 100% tails and 0% heads...he's thinking that there should now be 100 more heads than tails over the next 'long run' set of flips to 'even it out'

that's incorrect


what if it now goes 10mil heads to 10mil tails? that 100 no longer is significant...10mil heads to 10mil+100 tails is still 50/50 (49.99975% heads)...get it to billions etc and it's even closer

so no, it doesn't have to 'even out'...

and no matter how u look at it, there is no memory, so it's always 50/50 in any individual coinflip

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I like this explanation. It works well for the Monty Hall riddle too.




Follow [img]/images/graemlins/smile.gif[/img]

Re: Teaching an intelligent guy about a basic statistics concept
 

Its not really about statistics. Does your friend believe the coin or the universe keeps track of past results in a way that influences future results?

If he doesn't then he must immediately realise that he must be wrong. If he does believe the above then only experiments can hope to persuade him otherwise.

I'd guess his just confused about regressing to the mean. Pointing out to him that that doesn't mean 'making up' for the past but just means that the long term swamps the short term.

chez

Re: Teaching an intelligent guy about a basic statistics concept
 

I read through your post and your arguments matched up almost exactly with what I would have said to your friend in your same position.

He does not understand statistics. Or probability, if this is actually how he feels.

Re: Teaching an intelligent guy about a basic statistics concept
 

Please. Your friend is wrong. If you flip a coin, get heads and flip it again, everybody knows you're more likely to get heads.

http://www-stat.stanford.edu/~susan/...headswithJ.pdf

Re: Teaching an intelligent guy about a basic statistics concept
 

That sounds like a classic example of what's known as the "Gambler's Fallacy". If s/he is smart, s/he should be able to figure that out on his/her own.

S/he basically doesn't understand independent random events. This is an elementary concept and sounds like s/he doesn't truly have a grasp of probability and statistics. So I'd be skeptical of the "decent understanding of statistics" assumption you're making of your friend. No biggie, this is the sort of thing where human intuition doesn't serve us very well.

Re: Teaching an intelligent guy about a basic statistics concept
 

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http://www-stat.stanford.edu/~susan/...headswithJ.pdf

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To quote John McClane

"[censored] Calfornia"

Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

Re: Teaching an intelligent guy about a basic statistics concept
 

[ QUOTE ]
Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

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He was talking about using it in a real casino with a real bankroll. Actually, I don't think he had a concept of "bankroll" since he just assumed it couldn't fail.

Re: Teaching an intelligent guy about a basic statistics concept
 

I didn't read all the posts here so I apologize if this has already been brought up.

You are going to do a trial of 1000 coin flips of a fair coin. The statitiscal prediction is that there will be 500 each of heads and tails.

After 100 tosses, somehow heads has come up 100 times in a row. Using those results we now can recalculate that at the end of the trial, we now expect that there will be 450 tails and 550 heads.

The same concept can be applied to poker. Say you are a breakeven player and you start with a big bankroll. In the last year you have been very lucky and are up $5000. Even though you are a breakeven player, your total life winnings are now predicted to be $5000. You cannot expect bad luck over that time to bring you back to even since you are a breakeven player.

Your friend may be thinking along these lines in that over the course of those last 900 flips, we can't expect tails to "catch up" and reach the orginally predicted 500.

Re: Teaching an intelligent guy about a basic statistics concept
 

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S/he basically doesn't understand independent random events. This is an elementary concept and sounds like s/he doesn't truly have a grasp of probability and statistics. So I'd be skeptical of the "decent understanding of statistics" assumption you're making of your friend. No biggie, this is the sort of thing where human intuition doesn't serve us very well.

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imo the problem is that people don't understand what it means for events to even out in the long run. what the large law of numbers really says is that if you flip fair coins, the number of heads you get in n tosses is n/2 PLUS an error term that goes to infinity slower than sqrt(n).

Re: Teaching an intelligent guy about a basic statistics concept
 

[ QUOTE ]
imo the problem is that people don't understand what it means for events to even out in the long run. what the large law of numbers really says is that if you flip fair coins, the number of heads you get in n tosses is n/2 PLUS an error term that goes to infinity slower than sqrt(n).

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I think a simpler explanation is that people naturally want things to "even out" in the short term rather than the long term. "Infinite discrete distributions" is kinduva hard thing to understand.

Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

Now where's my grilled cheese w/ the Virgin Mary on it? I'm hungry.

Re: Teaching an intelligent guy about a basic statistics concept
 

Soon2bepro,

What math classes has this guy taken?

If you express the ratio of heads to tails as an infinite sequence you can show the addition or deletion of a any finite number of terms to the sequence won't change it's convergence (to .5).

Re: Teaching an intelligent guy about a basic statistics concept
 

[ QUOTE ]
Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

[/ QUOTE ]

This is a good point. I had a very good friend who was well trained in math and statistics (he became a chartered accountant in the end). We were talking about lotteries (powerball, 6/49 type). I mentioned that one could just pick numbers 1, 2, 3, 4, 5, 6. His initial reaction was, "that's stupid, what are the chances of that happening?". To which I answered, "just as likely as any other numbers". He had to think about it for a while before he convinced himself that I was right.

Re: Teaching an intelligent guy about a basic statistics concept
 

[ QUOTE ]
[ QUOTE ]
imo the problem is that people don't understand what it means for events to even out in the long run. what the large law of numbers really says is that if you flip fair coins, the number of heads you get in n tosses is n/2 PLUS an error term that goes to infinity slower than sqrt(n).

[/ QUOTE ]

I think a simpler explanation is that people naturally want things to "even out" in the short term rather than the long term. "Infinite discrete distributions" is kinduva hard thing to understand.

Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

Now where's my grilled cheese w/ the Virgin Mary on it? I'm hungry.

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obviously a careful explanation of the strong law of large numbers is a bit difficult to give to a layperson, but there's really no getting around it, because it is really what mathematicians mean by 'the long run'.

for example if i flip a coin a million times (a random walk on Z of 10^6 steps), my expected distance from the origin is about 2/pi*1000 ~ 600, even though my expected position is zero (equal heads and equal tails). mathematically, the law of large numbers says that the we get about half heads and half tails + a term that goes to infinity slower than n^(0.5+e) for ANY e>0 (this is related to the central limit theorem). this can still be a long ways from zero if n is large though.

qualitatively the important thing is that the mass of the distribution is clustered around {half tails, half heads} - this is the weak law of large numbers.

Re: Teaching an intelligent guy about a basic statistics concept
 

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Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

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He was talking about using it in a real casino with a real bankroll. Actually, I don't think he had a concept of "bankroll" since he just assumed it couldn't fail.

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Even with an unlimited bankroll and no betting limits, it's -EV.

A sum of -EV bets will never be positive. This is easy to see.

Re: Teaching an intelligent guy about a basic statistics concept
 

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obviously a careful explanation of the strong law of large numbers is a bit difficult to give to a layperson, but there's really no getting around it, because it is really what mathematicians mean by 'the long run'.

for example if i flip a coin a million times (a random walk on Z of 10^6 steps), my expected distance from the origin is about 2/pi*1000 ~ 600, even though my expected position is zero (equal heads and equal tails). mathematically, the law of large numbers says that the we get about half heads and half tails + a term that goes to infinity slower than n^(0.5+e) for ANY e>0 (this is related to the central limit theorem). this can still be a long ways from zero if n is large though.

qualitatively the important thing is that the mass of the distribution is clustered around {half tails, half heads} - this is the weak law of large numbers.

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I guess my thing is that people just have a hard time wrapping their heads around the whole infinity thing. I also prefer the integration of a more cognitive bias perspective/explanation.

Not that I have anything against the law of large numbers type of explanation. But, I mean, imagine if The Theory of Poker resembled The Theory of Gambling and Statistical Logic, all 8 copies would have sold as it's way too advanced for the average gambler. Sklansky's book is hard enough as it is.

I have a hard time seeing the OP's friend understanding such advanced concepts. I'd refer the student to the strong and weak laws of large numbers as an exercise. [img]/images/graemlins/smile.gif[/img]

Re: Teaching an intelligent guy about a basic statistics concept
 

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Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?


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No. Martingaling is the same thing as laying odds. Basically if you martingale with 2 bets a 50/50 payout game with vig, then you have a 75% chance to win, but you're getting paid less than a third of your total action, so it's unprofitable. If you go for 3 bets then you have an 87.5% chance to win, but you're getting paid less than laying those odds justify. That's the main thing. With an extremely large bankroll, your odds of winning increase, but still lag behind your money odds, and that's the only relevant thing.

As an aside, why would anyone waste there time making bets at a casino if they have unlimited money?

Re: Teaching an intelligent guy about a basic statistics concept
 

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Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

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He was talking about using it in a real casino with a real bankroll. Actually, I don't think he had a concept of "bankroll" since he just assumed it couldn't fail.

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Even with an unlimited bankroll and no betting limits, it's -EV.

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I think this can be debated.

Suppose the rules are as follows: heads gambler wins his bet, and tails he loses it. He can bet any amount at all and has an infinite bankroll. The probability of heads is slightly less than 0.5. The gambler initially bets $1, and doubles this bet every time he gets tails. After flipping heads he returns to a $1 bet.

Now consider a single game to end whenever he flips heads. What are the possible outcomes of a single game?

He has a 100% chance of winning $1 and a 0% chance of losing -$infinity.

Now it's true that if you do the math properly, apply limits etc., that the net EV of each game is -ve, however, the practical outcome of this game is that you will win $1 every time you play it. Which makes this a very strange -EV game.

Re: Teaching an intelligent guy about a basic statistics concept
 

Try an induction argument with n as the number of the trial.

Re: Teaching an intelligent guy about a basic statistics concept
 

Ask him to imagine that you're flipping a coin that has Heads on both sides. It comes up Heads ten times in a row. Does he think this coin is now less likely than before to come up Heads on the 11th toss?

Re: Teaching an intelligent guy about a basic statistics concept
 

Point your friend to the "law of averages" and "gamblers fallacy" wikipedia article. I am afraid you are vastly overestimating his or her knowledge of statistics.

Re: Teaching an intelligent guy about a basic statistics concept
 

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Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

[/ QUOTE ]

This is a good point. I had a very good friend who was well trained in math and statistics (he became a chartered accountant in the end). We were talking about lotteries (powerball, 6/49 type). I mentioned that one could just pick numbers 1, 2, 3, 4, 5, 6. His initial reaction was, "that's stupid, what are the chances of that happening?". To which I answered, "just as likely as any other numbers". He had to think about it for a while before he convinced himself that I was right.

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As it turns out, though, picking 1-2-3-4-5-6 is stupid, since you are punished for picking the same numbers as others.

Re: Teaching an intelligent guy about a basic statistics concept
 

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[ QUOTE ]
Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

[/ QUOTE ]

This is a good point. I had a very good friend who was well trained in math and statistics (he became a chartered accountant in the end). We were talking about lotteries (powerball, 6/49 type). I mentioned that one could just pick numbers 1, 2, 3, 4, 5, 6. His initial reaction was, "that's stupid, what are the chances of that happening?". To which I answered, "just as likely as any other numbers". He had to think about it for a while before he convinced himself that I was right.

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As it turns out, though, picking 1-2-3-4-5-6 is stupid, since you are punished for picking the same numbers as others.

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Umm, wouldn't it be less likely for someone else to pick the #'s 1-6?? Most people use the line of logic that above posters friend uses, and assume that something as simple as that number pattern would be less likely to occur.

Re: Teaching an intelligent guy about a basic statistics concept
 

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Please. Your friend is wrong. If you flip a coin, get heads and flip it again, everybody knows you're more likely to get heads.


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I meant a fair coin.

Re: Teaching an intelligent guy about a basic statistics concept
 

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To quote John McClane

"[censored] Calfornia"

Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

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This wasn't the topic, but just to clarify, martingale is only profitable if you have an infinite bankroll and an unlimited bet limit, but infinite is only a concept. however, no matter how large the number, if you have a limited bankroll or bet limit, it's not profitable. You can have a Googolplex dollars or betting limit, and still it won't be profitable. You need a truly infinite bankroll and no betting limit at all for it to be profitable. And as we know, infinite doesn't really exist.

Re: Teaching an intelligent guy about a basic statistics concept
 

[ QUOTE ]
[ QUOTE ]

To quote John McClane

"[censored] Calfornia"

Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

[/ QUOTE ]

This wasn't the topic, but just to clarify, martingale is only profitable if you have an infinite bankroll and an unlimited bet limit, but infinite is only a concept. however, no matter how large the number, if you have a limited bankroll or bet limit, it's not profitable. You can have a Googolplex dollars or betting limit, and still it won't be profitable. You need a truly infinite bankroll and no betting limit at all for it to be profitable. And as we know, infinite doesn't really exist.

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to be more accurate, its not profitable with an infinite bankroll because profiting implies having an increase. if you have an infinite bankroll then win 100million more dollars, you still just have an infinite bankroll. if you have an infinite bankroll is doesn't matter what strategy you take, as long as you bet finite amounts, you will always have an infinite bankroll.

so saying it works if you have an infinite bankroll is silly on multiple levels.

Re: Teaching an intelligent guy about a basic statistics concept
 

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Usually, the explanation of "each individual coin flip is even money" for either heads or tails assuming an unbiased/fair coin...and that T-T-T-T-T-T-T-T-T-T is as likely for a "random walk" as, say, H-T-T-H-T-T-H-T-H-H. All tails just looks funny; we are pattern recognizing machines after all, oftentimes regardless of any value in meaning.

[/ QUOTE ]

This is a good point. I had a very good friend who was well trained in math and statistics (he became a chartered accountant in the end). We were talking about lotteries (powerball, 6/49 type). I mentioned that one could just pick numbers 1, 2, 3, 4, 5, 6. His initial reaction was, "that's stupid, what are the chances of that happening?". To which I answered, "just as likely as any other numbers". He had to think about it for a while before he convinced himself that I was right.

[/ QUOTE ]

As it turns out, though, picking 1-2-3-4-5-6 is stupid, since you are punished for picking the same numbers as others.

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Umm, wouldn't it be less likely for someone else to pick the #'s 1-6?? Most people use the line of logic that above posters friend uses, and assume that something as simple as that number pattern would be less likely to occur.

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I'd be willing to make a large wager that the numbers 1-2-3-4-5-6 are picked more than the vast majority of arbitrary numbers.

Think of it this way, if there are say around 14 million combinations, you only need more than 1 in 14 million ticket buyers to think its funny/cool/smart to play 1-2-3-4-5-6 for it to become more frequently picked than an arbitrary number. (Wanna play the numbers from Lost? You are playing for a fraction of the jackpot now...)

For this reason, any number that you can come to by some simple mathematical system or pop-culture refernce, etc, is automatically worth less than a 'random' or seemingly arbitrary number with no special meaning.

Heck, if you really wanted to get the most +EV lottery numbers, you would probably write your own random number generator, and throw out any results that have the number 3, 7, 13.. and throw out any recognizable sequences (2-4-6-8-10-12) etc. But this is going way beyond the topic.

Re: Teaching an intelligent guy about a basic statistics concept
 

[ QUOTE ]
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[ QUOTE ]

To quote John McClane

"[censored] Calfornia"

Also, on the Martingale crap. Isn't it profitable- given no betting limit and an unlimited bankroll?

[/ QUOTE ]

This wasn't the topic, but just to clarify, martingale is only profitable if you have an infinite bankroll and an unlimited bet limit, but infinite is only a concept. however, no matter how large the number, if you have a limited bankroll or bet limit, it's not profitable. You can have a Googolplex dollars or betting limit, and still it won't be profitable. You need a truly infinite bankroll and no betting limit at all for it to be profitable. And as we know, infinite doesn't really exist.

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to be more accurate, its not profitable with an infinite bankroll because profiting implies having an increase. if you have an infinite bankroll then win 100million more dollars, you still just have an infinite bankroll. if you have an infinite bankroll is doesn't matter what strategy you take, as long as you bet finite amounts, you will always have an infinite bankroll.

so saying it works if you have an infinite bankroll is silly on multiple levels.

[/ QUOTE ]

I guess you're right, but I meant that you win more money than you lose, not that you increase your bankroll.

Re: Teaching an intelligent guy about a basic statistics concept
 

1.)What u need to stress to your friend is that the AVERAGE OBSERVED should head towards the TRUE AVERAGE. Not so much the TOTAL. As others have pointed out, if you flip ten heads in a row... your average is 100% ..Now imagine the next two flips.. on average, one heads, one tails, so now your average is 1/12, closer to 50% then 100%.. You still have 5 flips more than you would expect.

2.) Tell your friend to think of it like a coffee club. You pay $5 to get your first cup with a mug that you keep. Every cup there after costs you $0.25. So, after your first cup, your total and average is $5. Your second cup is .25, so your total is $5.25, but your average has dropped to $2.62. On your third cup your Total is $5.50, and your average is $1.83... So over time, that $5 investment that you made never goes away, but the average you spend on each cup keeps going down.

Re: Teaching an intelligent guy about a basic statistics concept
 

it seems like an explanation of conditional probability would end the discussion.

Re: Teaching an intelligent guy about a basic statistics concept
 

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I'd be willing to make a large wager that the numbers 1-2-3-4-5-6 are picked more than the vast majority of arbitrary numbers.

Think of it this way, if there are say around 14 million combinations, you only need more than 1 in 14 million ticket buyers to think its funny/cool/smart to play 1-2-3-4-5-6 for it to become more frequently picked than an arbitrary number. (Wanna play the numbers from Lost? You are playing for a fraction of the jackpot now...)

For this reason, any number that you can come to by some simple mathematical system or pop-culture refernce, etc, is automatically worth less than a 'random' or seemingly arbitrary number with no special meaning.

Heck, if you really wanted to get the most +EV lottery numbers, you would probably write your own random number generator, and throw out any results that have the number 3, 7, 13.. and throw out any recognizable sequences (2-4-6-8-10-12) etc. But this is going way beyond the topic.


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According to the lottery commission, the first 6 or seven numbers and the last 6 or 7 numbers are picked thousands of times per week. You want to avoid the numbers 1 through 12 and to a lesser extent, the numbers 1 through 31. You also want to avoid numbers that form a pattern on a lottery ticket, and I once read where even numbers are picked more often than odd numbers.

Re: Teaching an intelligent guy about a basic statistics concept
 

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Please. Your friend is wrong. If you flip a coin, get heads and flip it again, everybody knows you're more likely to get heads.


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I meant a fair coin.

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If you had read the paper you would know it refers to a fair coin. To sum it up, if you try flipping a fair coin as fair (vigorously) as possible, it's about 51% to come up the way it started.