Is it true that coin flips are always 50-50?

[Peng]

Need help! My friend thinks coin flips are 50-50!

Alright, I've been trying to convince my friend that the outcomes of a coin flip x times in a row affects the x+1'th time. If you flip a coin 4 times and they are all heads, the 5th time is more likely to be a tail because if the coin is even, over time there should be as many tails as there are heads. I even tried to prove it to him using math:
P(5 heads in a row) = .5^5 = 0.03125
P(4 heads in a row) = .5^4 = 0.0625

Which directly translates into: 5 heads in a row is less likely to happen than just 4 heads in a row so if you flip a coin and get 4 heads, the 5th time is more likely to be a tail.

My friend just doesn't understand this, he's saying that even when I'm calculating the probabilities, I'm using .5 for each flip so the probability should be .5 every time, but he's obviously wrong, because the probabilities are different! Please help me convince him that my way is right...after all why does he think when they do coin flips, they do best out of 3 times?

 

[NeutronStar]

I have a feeling this is going to be an interesting thread.

P.S. If you remove the insulting adjective before the word friend in your title you might get more consideration for reasonable replies.

 

[Peng]

How do I do that?

Never mind...figured it out.

 

[Hurkyl]

Try computing P(4 heads followed by a tails). How does it compare to P(4 heads followed by a heads)?

 

[Peng]

They're equal, but how does that prove that what I said is true?

 

[Hurkyl]

It doesn't: it supports your friends position.

 

[Chronos]

Statistically, the odds of n number of consecutive results, given a 50-50 outcome probability, is 2^(n-1). Don't bet with your stupid friend unless you don't mind getting sheared like a cashmir goat.

 

[matt grime]

I didn't realize it was a rule that you must toss a coin "best of 3".

 

[Peng]

Waaaait wait a second here. So you all are saying that my friend is right?
But the probabilities say that if you flip a coin 6 times, you should get about 3 heads and 3 tails, so if you flip it 5 times and get 5 heads the 6th time should almost deffinately be a tail.

 

[chroot]

Your friend is right, and you are wrong, without question. The coin flips are entirely independent. Everytime you flip a coin, you have a 50-50 chance of getting both heads or tails, no matter what happened before.

But the probabilities say that if you flip a coin 6 times, you should get about 3 heads and 3 tails, so if you flip it 5 times and get 5 heads the 6th time should almost deffinately be a tail.

Wrong, without question. The probability is that half the time you flip a coin, you'll get heads, and half the time you'll get tails, on average. That's all.

Your logic is actually what suckers people into playing games like roulette. In casinos, roulette tables have a big flashy sign above them that lists the results of the last ten spins of the wheel. Passing gamblers see that a wheel has spun, say, ten blacks in a row, and they can't help but think there's a higher chance to spin red next. That false hope helps make the casino richer.

To illustrate your thought process, let's look at the possible results of spining a roulette wheel twice:

BB
BR
RB
RR

Note that there are two possible sequences with one black and one red spin, but only one each of all black or all red. This leads people -- including yourself -- to mistakenly think that sequences that have an equal mixture of reds and blacks are more likely than sequences of all red or all black. This is not true!

Now let's look at the possibilities with 10 spins:

All blacks (1 possibility)
Some mixture of blacks and reds (1,022 possibilities)
All reds (1 possibility)

Do you see what's happening here? There are 1,024 possibilites, only two of which are all black or all red. That leads people -- including yourself -- into thinking that a sequence of all reds or all blacks is highly improbable. It's not! The sequence BBBBBBBBBB is no less probable than the sequence RBRRBRBRBB. Sure, one looks more "random," but, in fact, both sequences have exactly the same probability. Every possible sequence of spins has a 1/1,024 chance of happening.

Richard Feynman one explained this misconception to a class by announcing that he had just come in from the parking lot, where he saw a car with the license plate ARW457. He expressed his amazement that, of all the millions and millions of license plates in the state of California, he had seen that one!

Sometimes people refer to "the law of large numbers" when dealing with probabilities. Only if you flip the coin a large number of times can you be certain of getting 50% heads and 50% tails. If you flip it just once, obviously you don't -- you get either 100% heads or 100% tails. Only if you flip the coin an infinite number of times, in fact, are you guaranteed of getting 50% heads and 50% tails.

Your friend is the smart one, Peng, and you are the one who is "stupid." Coin flips are independent. No matter what you flipped in the past, the probability of each flip is 50-50. The universe does not conspire to make coins work differently from one flip to the next. Even if you flipped a coin ten billion times and it came up heads every time, there is no greater chance of it being tails the next. Your friend is a smart one -- stick around, and you might learn something from him/her.

- Warren

 

[chroot]

I even tried to prove it to him using math:

P(5 heads in a row) = .5^5 = 0.03125
P(4 heads in a row) = .5^4 = 0.0625

Which directly translates into: 5 heads in a row is less likely to happen than just 4 heads in a row

No, it doesn't. What you actually compared was the probability of any individual 5-flip sequence to the probability of any individual 4-flip sequence. Since there are more possible 5-flip sequences than possible 4-flip sequences, it's clear that each invidual sequence's probability should be smaller.

Why don't you instead compare these two sequences:

P(HHHHH) = 0.5^5 = 0.03125
P(HHHHT) = 0.5^5 = 0.03125

- Warren

 

[Hurkyl]

Here are some other ways of internalizing this fact:

You've flipped 5 coins, and got 5 heads. Now, you are performing an experiment with one coin flip -- and you know that one coin flip has a 50% chance of being heads.

getting 5 heads and 1 tail is more likely than getting 6 heads, because there are more ways to get 5 heads and a tail:

HHHHHT
HHHHTH
HHHTHH
HHTHHH
HTHHHH
THHHHH

vs

HHHHHH

However, because you started with 5 heads, you've ruled out all of those other possibilities that made 5 heads and a tail more likely than 6 heads.


And to help internalize your mistake (which, as chroot mentioned, is common), you rationalized that on 6 coin flips that you are most likely to get 3 heads and 3 tails... but that's absolutely impossible once you've seen 5 heads, so you should know that there was a kink in your reasoning.

 

[courtrigrad]

yes. coin tossing contains both markov and martingale properties, and are entirely independent from one another. This same principle works with payoff, volatility and drift terms. Your expected payoff will be what you currenty have up to $$P_n_-1$$ days will just be the current amount you have.

 

[Galileo]

Wouldn't it be nice if Peng was right?

I would toss a coin 5 times till I got 4 heads in a row on the first 4 throws of the sequence.
Then I'd walk up to my brother and say: "Wanna bet 10 euro's that this coin toss gives head?"
The odds would be greatly in my favour.

 

[Ethereal]

I found two Wikipedia articles on this:

http://en.wikipedia.org/wiki/Inverse_gamblers_fallacy
http://en.wikipedia.org/wiki/Gambler's_fallacy

 

[NeutronStar]

A Computer Simulation Program

I never trust mathematics so I had to prove to myself that Peng was wrong by writing a simulation program.

Years ago, in 1996 I had a bit of a problem believing this one:

http://www.cut-the-knot.com/hall.shtml"

After writing a BASIC program to simulate the doors I could see that Marilyn was right. Of course, that problem also loans itself to intuitive insight by simply imagining that there are thousands of doors.

In any case, I know that coin tosses must always be 50/50 because there's no way that history can change how a coin behaves. None the less, the Monty Python Dilemma does seem to suggest that since we have more information this might somehow have an effect on the probabilities. So I wrote the following Visual Basic program to toss the coin and keep track of the results.

The program pasted below is hard-wired to only tally up the heads or tails that come up after 4 heads in a row. It also only does this for 1000 tosses. I actually ran the program to toss millions of times, and played with various numbers of heads in a row. The results were always 50/50 after millions of tosses.

Anyhow, for anyone who's interested in BASIC programming here's the program code. It uses a form with a Flip Coin button called cmdFlip, and has three text boxes named, txtTallyHeads, txtTallyTails, and txtPaper where I print out the actual results of tossing a coin 1000 times. I place underscores after every occurrence of N-heads in a row so they can easily be found in the print out. I've also attached a small picture of the output. I had to clip it down to fit into PF's 400 pixel limit for GIFs.

It seems that Peng's friend is right. Which I figured had to be the case since coins have to be 50/50 no matter what. But in light of the Monty Python Dilemma I thought I'd just simulate it to be sure. Sorry Peng, you were wrong.

Here's the code for the VB program: (be sure to scroll down to see the last part)
A clipped image of the output of 1000 tosses is also attached.

Option Explicit
Dim Toss As Double: Rem For the coin tossing loop.
Dim Flip As Double: Rem Just a random flip of the coin
Dim Coin As String: Rem Keeps track of how the coin landed
Dim N As Integer: Rem The number of heads in a row to test.
Dim CountHeads As Integer: Rem Counts heads in a row.
Dim TallyHeads As Integer: Rem Tallys # of heads after N-heads
Dim TallyTails As Integer: Rem Tallys # of tails after N-heads

Private Sub cmdFlip_Click()
Rem This routine actually flips the coins
TallyHeads = 0
TallyTails = 0
N = 4
txtPaper.Text = "": Rem clear the paper
For Toss = 1 To 1000
Flip = Rnd(1)
If Flip < 0.5 Then Coin = "T" Else Coin = "H"
txtPaper.Text = txtPaper.Text + Coin
If CountHeads = N Then Call Tally_Toss
Call Count_Heads
Next Toss
End Sub

Public Sub Count_Heads()
Rem This routine counts N heads in a row
If Coin = "H" Then CountHeads = CountHeads + 1
If Coin = "T" Then CountHeads = 0
End Sub

Public Sub Tally_Toss()
Rem This routine tallies the heads or tails after N-heads in a row.
    If Coin = "H" Then TallyHeads = TallyHeads + 1
    If Coin = "T" Then TallyTails = TallyTails + 1
    txtTallyHeads = Str(TallyHeads)
    txtTallyTails = Str(TallyTails)
    txtPaper.Text = txtPaper.Text + "_____": Rem Just a marker
    CountHeads = 0
End Sub

 

[cepheid]

It's Monty HALL...the host of the game show Let's Make a Deal. And I don't understand how it cast into doubt your understanding of whether coin tosses are 50/50 or not. The situations don't seem related in any way...why bring it up?

 

[matt grime]

And why say marilyn was right when she got it wrong because she didn't know anything about conditional probablity? I trust you built the basic program to allow for the fact that the door they open is KNOWN to not have the car behind it?

 

[HallsofIvy]

And why say marilyn was right when she got it wrong because she didn't know anything about conditional probablity? I trust you built the basic program to allow for the fact that the door they open is KNOWN to not have the car behind it?

A good reason to say Marilyn vos Savant was right is that she was!

I have no idea whether she did or did not know anything about conditional probability but her answer was correct. The question was "Suppose you choose one of three doors, knowing that there is a car behind one of them, equally likely to behind anyone of the three doors. The game show host (Monty Hall) who knows which door the car is behind opens a door showing the car is not there. Would the person improve his/her chances of winning the car by changing his/her choice?

Marilyn vos Savant did not use conditional probability to answer: you did what mathematicians often do to think about a problem intially- look at an extreme case. She said "suppose there were 1000 door, with a car behind one of them. You choose one, the game show host opens 998 of the doors showing no car behind them. In other words, there are now two doors, your choice and one other, one having the car behind it. Would you change? You bet you would! Monty Hall has completely changed the odds, using his superior knowledge.
Of course, one can apply conditional probability to show that same result. I remember seeing this problem about 15 years ago as an exercise, in chapter one of an introductory probability book.

It is an interesting exercise to see what happens if Monty Hall does NOT know which door the car is behind but opens doors at random. You can use conditional probability to show that, in that case, given that the door he opens HAPPENS not to have the car behind it, there is no advantage to changing.

 

[Hurkyl]

IIRC, Marylin first gave the wrong answer, then later gave the right answer.

 

[Ethereal]

IIRC, Marylin first gave the wrong answer, then later gave the right answer.

Can you post or at least direct me to her answers? I know she got it partially right the first time but I wasn't sure if there was a second time.

 

[matt grime]

There is a website devoted to Marilyn's wrong answers, try googling for it.

 

[NeutronStar]

It's Monty HALL...the host of the game show Let's Make a Deal. And I don't understand how it cast into doubt your understanding of whether coin tosses are 50/50 or not. The situations don't seem related in any way...why bring it up?

I would argue that they are related. Or at the very least they certainly seem to be related. In both cases you are given addition information in the middle of the game.

In the case of picking doors that information is useful, in the case of flipping coins the historical information is not useful.

But it does seem that most people intuitively feel like it should be useful. Sites like Ethereal gave earlier on Wikipedeia clearly state that it is a very common misconception.

I think it does relate to The Monty Hall Dilemma. Imagine a game show where the host is going to flip a coin. So the contestant picks heads. Then the host starts flipping the coin, but tells the contestant that only the fifth flip will count. The coin lands heads up on the first four flips. Then the host asks the contestant if he or she would like to change their choice before the fifth flip. How many people would change their choice?

It does kind of parallel the Monty Hall Dilemma. The host has given the contestant additional information concerning the history of that particular coin being flipped at that particular time. The only difference here is that this additional information is totally useless.

Yet, many gamblers keep track in their minds of the history of the events of the games that they play. They make decisions based on that history all the time. In fact, there are successful gamblers who would go to their grave claiming that it helps them to win! Of course, in card games history does help because cards that are dead can't be in a hand, and that does change probabilities. But in things like craps or roulette history doesn't help at all.

I think there's something to be learned by comparing tossing coins to the Monty Hall Dilemma. Different kinds of information make a difference. In some cases additional information (even historical information) can help, in other cases, it isn't any help at all.

So I would argue that there does seem to be a relation between flipping coins and picking doors. It's just that the relationship is not exact. They are similar, but different.

I certainly disagree with your statement; "The situations don't seem related in any way...why bring it up?" They are related. They just aren't identical.

Another very enlightening situation that is also related is that of the Truel. A gunfight between three people. It might not seem to be related either but it is related in a totally different way.

http://www.maa.org/mathland/mathtrek_1_26_98.html"

 

[russ_watters]

I would argue that they are related. Or at the very least they certainly seem to be related. In both cases you are given addition information in the middle of the game.

In the case of picking doors that information is useful, in the case of flipping coins the historical information is not useful.

But it does seem that most people intuitively feel like it should be useful. Sites like Ethereal gave earlier on Wikipedeia clearly state that it is a very common misconception.

I think it does relate to The Monty Hall Dilemma. Imagine a game show where the host is going to flip a coin. So the contestant picks heads. Then the host starts flipping the coin, but tells the contestant that only the fifth flip will count. The coin lands heads up on the first four flips. Then the host asks the contestant if he or she would like to change their choice before the fifth flip. How many people would change their choice?

It does kind of parallel the Monty Hall Dilemma. The host has given the contestant additional information concerning the history of that particular coin being flipped at that particular time. The only difference here is that this additional information is totally useless. [emphasis added]

Here's how I like to think about it: how does the coin know what it came up with in the first 4 flips? (A: it doesn't, so that can't affect the next flip). Always ask yourself that question and you'll always come up with the right answer.

-How does the lottery know its never picked your number?
-How does the Roulette wheel know it hit black the last 3 times in a row?
-How does the blackjack deck know it just dropped 2 aces in a row?

(that last one is a trick...)

 

[robert Ihnot]

russ waters: Here's how I like to think about it: how does the coin know what it came up with in the first 4 flips? (A: it doesn't, so that can't affect the next flip). Always ask yourself that question and you'll always come up with the right answer.

What if you picked a coin from a coin box and do four flips heads. Then you replace the coin in the box, and again pick a coin from the box. Does it matter which coin you have now removed from the box?

 

[Eyesaw]

If we knew all the forces involved in the coin flip, shouldn't the probability of it
landing heads or tails more like 100%? I thought the 50/50 figure is symbolic of the incomplete information we have for predetermining an outcome with two possible results and not a real figure? However, if we flipped a coin a million times and it all landed heads, I'd think the chance of the next flip being
tails would be 1 in a million. The fact that it landed heads all those times would
tell us the conditions of the flip greatly favoured heads.

 

[matt grime]

Or that the coin weren't fair. But this isn't a question about real life, it is about mathematical models where we assume independence. These then model real life quite well, but aren't always applicable. See Hypothesis Testing, for example. In the case you describe we can safely reject the assumption that {the coin is fair and the tosses independent}.

 

[Eyesaw]

Or that the coin weren't fair. But this isn't a question about real life, it is about mathematical models where we assume independence. These then model real life quite well, but aren't always applicable. See Hypothesis Testing, for example. In the case you describe we can safely reject the assumption that {the coin is fair and the tosses independent}.

Anyone who has watched those Asians juggling knives on TV would disagree
with your assumption of independence and the correctness of the 50/50 model
for coin tosses.

 

[matt grime]

So, wait, someone catching a knife, presumably not at random, implies that I should disregard the notion that it is reasonable to assume independent and even odds on tossing some coin in my pocket? Wow, that's possibly the most tongue in cheekly ironic post I've seen.

The ball in roulette has a 1/33 (or 1/34 for the US) chance of landing in any position is the model. Spins are independent, we'll say. So the casinos are going to go bust soon, then, are they, because of a juggling trick?

 

[Eyesaw]

So, wait, someone catching a knife, presumably not at random, implies that I should disregard the notion that it is reasonable to assume independent and even odds on tossing some coin in my pocket? Wow, that's possibly the most tongue in cheekly ironic post I've seen.

The ball in roulette has a 1/33 (or 1/34 for the US) chance of landing in any position is the model. Spins are independent, we'll say. So the casinos are going to go bust soon, then, are they, because of a juggling trick?

A coin flip and a knife flip are the same mathematically. Both have two possible outcomes. I used the knife flip as example because it is easier to control the flip of a knife than the flip of a coin. The point was that a person can determine with 100% probability each and every knife flip and since a coin flip is just the same type of action on a smaller scale, there is no such thing as a 50/50 probability for the outcome of a flipped coin. So of course that is a very unreasonable assumption you are making.

As for the roulette spin, have you heard of the Eudaemon Pie? If the casinos allowed a person to use a computer system to calculate the landing of the ball, they would go bust very quickly. And if they mechanized the spinning of the wheel as well as the tossing of the ball, the 1/33 chance of landing on any number would immediately be found false as well.

Yes, when the system gets complicated, like say you flipped the coin or knife a 100 meters in the air through a hurricane, the outcome becomes effectively unpredictable but then in that case, a ratio of 30/70, 40/60, 20/80 or whatever besides 0/100 is just as valid as 50/50 since we can always claim that after more and more trials, the number should get closer to our chosen ratio if it wasn't so after some number of flips. In other words, the 50/50 probability is pretty meaningless and doesn't aid in identifying nor explaining the causes of the outcome, which are the physical forces involved in the event.

Another beef I have with that is that it kind of misleads one to think of probability as part of the forces that control the outcome of the flips. As if there is some probability engine in the universe that goes to normalize the ratio of heads to tails to 50/50 after a "large" number of trials.

 

[matt grime]

No mathematician thinks of probability as causal in the sense you so describe.

You realize you've now implicitly said that "everything with two possible outcomes is mathematically the same"? That is a fairly meaningless comment.

I am making no unreasonable assumptions: if the 50-50 long term behaviour is false for a given coin in a given situation then it is not a good model. Nor am I attempting to explain the reasons why a coin falls head or tail, or lands on its edge and gets stuck in a crack. Probability is a useful way of modelling such things. As a simple matter, flipping a coin 2 feet in the air is sufficient to use a probablistic argument to describe its behaviour. Ever heard of Buffon's Needle and Monte Carlo methods?

Why do you have a problem with probability failing to do things it doesn't claim to do?

A person would not determine with 100% probability the outcome of each flip, by the way. He woudl determine with 100% accuracy. Not that we believe that to be possible in a non-linear model, or a quantum mechanical one. We do not claim that with 50% probability we can determine the outcome, which is what you are implying there.

 

[Eyesaw]

No mathematician thinks of probability as causal in the sense you so describe.

You realize you've now implicitly said that "everything with two possible outcomes is mathematically the same"? That is a fairly meaningless comment.

I am making no unreasonable assumptions: if the 50-50 long term behaviour is false for a given coin in a given situation then it is not a good model. Nor am I attempting to explain the reasons why a coin falls head or tail, or lands on its edge and gets stuck in a crack. Probability is a useful way of modelling such things. As a simple matter, flipping a coin 2 feet in the air is sufficient to use a probablistic argument to describe its behaviour. Ever heard of Buffon's Needle and Monte Carlo methods?

Why do you have a problem with probability failing to do things it doesn't claim to do?

A person would not determine with 100% probability the outcome of each flip, by the way. He woudl determine with 100% accuracy. Not that we believe that to be possible in a non-linear model, or a quantum mechanical one. We do not claim that with 50% probability we can determine the outcome, which is what you are implying there.

I agree with what you are saying and my previous posts weren't a direct response to your posts really though I really don't see how probability is very useful for describing a coin toss.For like a 6 sided dice that have 5 numbers the same for example it would be useful.

I was reducing the knife throw to just two possible outcomes to make an analogy with the coin toss. The 50/50 hypothesis for the coin toss is arrived at the same way after all.

 

[master_coda]

I agree with what you are saying and my previous posts weren't a direct response to your posts really though I really don't see how probability is very useful for describing a coin toss.

The funny thing about this is that most of the time I see mathematicians talking about coin tosses, they don't actually care about coin tosses. They care about a probability model where you have exactly two possible outcomes that have the same probability of occurring. Coin tosses are just used as a physical example of such a model to make things easier to understand for people who have a hard time understanding abstractions.

Seriously; who wants to waste their time modelling coin tosses?

 

[matt grime]

I was reducing the knife throw to just two possible outcomes to make an analogy with the coin toss. The 50/50 hypothesis for the coin toss is arrived at the same way after all.

The knife is thrown in a carefully rehearsed pattern by a skilled performer. The dynamics of the knife rotating are much different from the coin. The knife is caught most definitely not at random by the same skilled perfomer.

The coin is thrown in the air without any great care, the forces of the toss will usually make it spin faster than human perception allows us to differentiate between sides, and is either plucked out of the air with no great care again, or allowed to fall onto some surface.

Yeah, they're the same.

The reason we assign a probability of 1/2 to it landing heads is that, in the long run, that is what our experience tells us the proportion of heads to tosses will approach.

It is not the fact that there are two outcomes, it is the fact that there is no reasonable assumption as to why one should be preferred over the other - they are equally likely. Completely unlike the probability that a knife juggler loses a finger.

If you don't believe this to be reasonable then please do look up the Buffon Needle experiment where probability can and has been used to calculate the value of pi to a reasonable accuracy.

 

[so-crates]

A non-mathematical way of looking at it:

Does the outcome of a series of coin flips affect the probability of getting heads/tails on the next flip? If it does, how long would you have to wait before the probabilites "reset" themselves to 50%? An hour? A day? A year?

Does the outcomes of *your* coin tosses affect the outcomes of *my* coin tosses? If you believe that the outcome of previous coin tosses affect the outcome of the future coin tosses, then why not? If I get 5 heads in a row, the probabiliti of you getting heads should be a lot lower, because the probabilit that both of us will get 6 heads is a lot lower, right?

Does the number of times people flipped heads in entire human history affect *your* odds of getting heads/tails on a given flip? If you believe that the outcome of previous coin tosses affect the outcome of the future coin tosses, then why not? Let's say the running total for the entire human race is now 51% heads. Shouldn't you have a higher chance of getting tails?


I hope I not confusing you, the answer is NO, the trials are what statisticians like to call "mutually exclusive events", and the probability is always 50/50 (well some claim you get heads slightly more often, due to the aerodynamics and weight distrubution of the coin)