 # Randomness, coin flipping and probability

I have a coin, one side is heads, the other side is tails. If I flip this coin 10 times, I get the following results:

H H H H H H H H H T

Doesn’t this mean that the probabilty of getting tails increases, with each flip that comes up heads? My own sense of probability tells me that this is true, but mathematically it is probably false.

If I flip a coin an infinite number of times, can I guarentee that eventually it will come up heads? The chances of it never coming up heads are so vanishingly remote, that we can assume that it certainly will happen?

These are questions I have often wondered about, and as we have quite a few mathematically minded dopers, I thought you could help me understand the principles behind it.

(The above is assuming that flipping the coin produces a truly random result.)

Nope. Although people ‘feel’ a given outcome is ‘due’ there is no basis in reality for it.

Each coin flip is completely independent of the previous or following coin flips. Each time you flip the coin you have an (ideally) 50/50 chance of it being heads or tails. If you flip 100 heads in a row there is no better chance for tails too come up on flip 101. It is still a 50/50 proposition.

Here’s one that will fly in the face of your gut instinct.

Playing the lottery with the three following combinations have exactly the same chance of winning:

01 01 01 01 01

01 02 03 04 05

52 47 26 12 33

Knowing that would you ever dare to put all ‘01s’ on your lottery ticket?

i read this somewhere but it is a fact that the heads side of a coin is slightly heavier than the tails side and if you flipped a coin 100 times you would get about 45 heads and 55 tails besides the fact that is is mostly chance in the flip

probability is hard to do with a coin because of these reasons -

• you have to start on the same side of the flip everytime
• you have to make sure that there is no wind at all
• you have to make sure there is an equal weight on both sides
• and some other reasons that i cant come up with right now

No. For that to be true, the coin would need some sort of memory mechanism, in order to decide how future probability should be influenced by the past.

It might mean the opposite. If you watch a bunch of coin flips and most of them come up heads, one possibility is that it’s not really a fair coin – maybe heads are more likely than tails.

Since you can’t actually flip a coin an infinite number of times, it’s a little hard to say whether this is a meaningful question. It is correct that with enough flips you can reduce the chance of no heads to as small a value as you care to name.

If you are using a REAL coin and get 9 heads and then a tail, the odds are that the coin is not “fair”… and I’d bet on the next flip also giving heads.

If the coin is absolutely fair, then the chance of the next flip coming up heads is exactly 50%, ditto the chance of tails.

If you have trouble with that, think of it this way: think of a “trial” as being ten flips of the coin. Now do a hundred trials – so you’ve got 1000 flips of the coin.

A few (but not many) of the trials may come up nine or ten heads. A few (but not many) will come up nine or ten tails. Most will come up with about half heads, half tails – like 4 and 6, or 5 and 5.

In any case, if the coin is fair, the prior results have no bearing on the next results.

That’s something else I have wondered. People use things like their birth date, age, etc. But they have equal chance with 1 1 1 1 1.

I will expand on the coin thing a little, and hopefully make my question clearer:

1. Assume that if I flip the coin an infinite number of times, it will come up tails eventually.

2. Let the number of times needed to flip it, before tails comes up = x

Now, x could equal 0, 1, 50 or any number. That is the nature of randomness.

Example 1: H H H H T

for this, x=5

Example 2: T

for this, x=1

Now, for any given sequence of flips, x will have a definite number. From this, can we not say that the probability of flipping tails increases? Maybe I’m just confusing myself, but it seems that as we approach x, there is more chance of flipping a tail. This is because x is probably a low number, like 1, 3, 4 etc.

samarm said:

I do not understand this part of your post. Could you explain further?

Interestingly enough I was standing in a newsagent’s queue with a friend who was lodging his lotto entry and we were talking about this. I said to him that it would be a good idea to take things that no-one would take like 1 2 3 4 5 6 to ensure that you get the whole pool. The newsagent chimed in and said that he had several customers at his agency alone who took this combination. He said that lots of people take lots of the intuitively “unlikely” combinations. The most winners ever in an Australian lotto was on 24 July 1993 when 150 people shared the \$2.3m first division prize. Usually only 1 or 2 people would win. I haven’t yet been able to find the numbers drawn.

What you’re asking for is the probability of getting a tail within k flips. That’s a very different problem from what you posted in the OP. While the probability of getting a tail on any given flip is constant, the probability of getting no tails in k flips does decrease. I’d use the geometric distribution to solve this problem.

As far as an infinite sequence of coin flips goes, that’s a bit of a can of worms for some people. There is a single right answer, but people don’t like to accept it. If you were to flip a coin an infinite number of times, the probability of not getting any tails would be exactly zero. The probability of getting any specific sequence of flips is also zero. And yet, one of them happened…

Sorry guys, that last post was in response to Whack-a-mole. I’m having BIG connection problems at moment. I’ll try my best to keep up with the thread, and I am reading all the responses, rest assured.

I don’t think this is true? Don’t the lotteries have one ping-pong ball per number? There are certainly not 52*5 ping pong balls in that little popcorn popper dealy.

Maybe they run lotteries differently where you are, but with most state lotteries I’ve seen the first combination would be infinitely less likely. There is usually only one ball with each number.

C K Dexter said:

<< A few (but not many) of the trials may come up nine or ten heads. A few (but not many) will come up nine or ten tails. Most will come up with about half heads, half tails – like 4 and 6, or 5 and 5. >>

So, going back to my previous post, x is most likely to be a small number, like 1 or 2.

Therefore, it is less likely to require 10 flips to get a tail, than it is to require 2 flips. Sorry, I think it may just be me, but to my mind that indicates that the more flips that come up heads, the nearer we get to x, the more chance of the next flip being tails.

ultrafilter said:

<< As far as an infinite sequence of coin flips goes, that’s a bit of a can of worms for some people. There is a single right answer, but people don’t like to accept it. If you were to flip a coin an infinite number of times, the probability of not getting any tails would be exactly zero. The probability of getting any specific sequence of flips is also zero. And yet, one of them happened…>>

Exactly zero? Even through there is a 50/50 chance? Isn’t it even remotely possible that it would always be tails?

Oh yes, it is possible. It just has probability zero. That’s the whole “paradox”, if you will. It’s not really paradoxical, but fairly counterintuitive.

As far as flipping a coin until it comes up tails, you don’t expect to do it more than twice. The probability of not getting any tails in n flips is (1/2)[sup]n+1[/sup], based on the geometric distribution.

samarm, I believe you’re confusing two separate issues:

Say you’ve flipping a coin a bunch of times. A couple of questions we could ask:

1. What is the probability of the 1000th flip being a tail?

2. What is the probability of any of the first 1000 flips being a tail?

The answer to the first question is 50% (this is regardless of what particular flip we’re considering).

The answer to the second question is very close to 100% (1 - 1/2[sup]1000[/sup], actually).

The more times you flip, the more likely a tails has occured at some point–which is only common sense, since if it has a 50% chance of occuring at any given time, it likely won’t take long before it happens. However, the chance of any particular flip being tails always remains 50%. Does that clear it up any?

samarm, the probability of getting either heads or tails is the exact same every time. Each trial is independent of any other trial coming before or after; the probablity never changes. Even if you flipped a coin 100 times and it landed tails every time(assuming it isn’t a biased coin) the probability of the next trial coming up heads is no larger or smaller.

Now the probabilty of getting nine tails in a row is quite small

(1/2)^9

But that is the probability of getting a certain number in a row…

The probability that the next coin comes up heads is still (1/2) because it doesn’t matter what happened before that.

However, if you were to start from the beginning and were going for 10 in a row the probabilty that you won’t make it gets very large.

P(You don’t get ten in a row)= 1 - (1/2)^10

Each successful trial makes the probability above smaller until the last one (where there have already been 9 in a row) where the probability is (1/2).

But that is not the same as…

P(The next coin coming up heads)= (1/2)

Clear? The probabilities of independent events occuring together do not affect the probabilty of each independent event.

OK, the question about the probability increases with the number of flips - I think I got it now. Thanks for the explanations.

The thing about the infinite number of flips, I’m still intrigued by. However the explanation is probably beyond me, so I won’t press for more detail.

Thanks again to all.

The probability of getting heads with any single flipping of a coin is .5

The probability of getting heads within two coin flippings is 1 - (.5 * .5) = .75

The probability of getting heads within three flips is 1 - (.5 * .5 * .5) = .875

Therefore, if you flip the coin more, the chances that you will get heads increases, so if you flip the coin an infinite number of times, you will eventually get a heads. However, the chance that any individual coin will be heads is always .5 and does not depend on what the previous flips were.

If the previous flips were HHH, then the chance that the next result will be H is still .5. Likewise, if the previous flips were TTT, the chance that the next result will be H is still .5.

Hmm…I understand the math of this, but I’ve always had trouble rationalizing it intuitively (as with a lot of probability theory).

I understand that the coin has no memory, so that for each flip there is a .5 probability of getting H or T (assuming a perfectly fair coin), regardless of previous outcomes.

Also, I understand that even if you’ve flipped this perfectly fair coin and got 9 Hs, the probability on your 10th flip of getting an H is still 0.5.

I understand mathematically that if you decide ahead of time that you want 10 Hs in a row, the probability of that is (.5)[sup]10[/sup] - i.e., something small, much < .5.

I just can’t rationalize this intuitively - I’m grasping for a visceral feel for it but coming up short. My problem with it is precisely because each flip of the coin is independent of the other flips, and the coin has no “memory”. Why should 10 Hs in a row be any less probably than any other sequence? The coin has no preference. I had the same trouble in high school math, with problems like the probability of rolling a 3 and a 3 with two dice, compared to, say, a 3 and a 4. I know there’s only one way of getting 3-3, and 2 ways of getting 3-4, but intuitively the dice shouldn’t be able to discriminate because the 3s happen to look the same. I guess there are two things to be gleaned from this: (a) intuition is not always a good guide in math, especially probability and (b) I’m not able to articulate very well what I don’t like about these answers.

Yes, I have actually tried experiments with coins and dice, both real and simulated, to convince myself of these numbers. Still doesn’t quite feel right, though. There seems to be something essentially “ungraspable” about probability, whereby you just have to trust the formulas.