Law of Averages and Coin Tosses

Factual Questions
1

Obviously, my knowledge of probability is limited, but why is it that if I flip a coin enough, I should come out with about half head and half tails? Isn’t the chance (disregarding factors such as air currents, weight imbalance and varying initial velocity) for any individual coin toss to be heads .5? If not, why?

To put this in context, a few years ago my cousin and I were playing a game which involved flipping coins. After several tails in a row he seems to become more and more confident that it would be heads. Why? Do the gods of averages mystically compell the coin to land a certain way?

(Note: I’m sure I said some ignorant things in the post above. Honestly though, I want to know.)

2

Since each coin toss is independent of the previous ones, a row of tails gives no indication that a head is coming soon.

If the probability of getting a head on a given toss is .5, then the expected proportion of heads is .5.

I’m not going to say anything more just yet, because my brain is sore from doing math homework, and I have a good chance of confusing the issue.

3

The law of averages doesn’t mean that you can’t toss heads 20 times in a row. It just means that if you toss the coin a thousand times you should come out with nearly 50% heads and 50% tails (give or take a few). The more you toss the closer the heads/tails averages should converge on 50%.

After seeing 5 heads tossed in a row people tend to think tails are due and will bet accordingly. However this feeling is artificial and bears no relevance to reality. All things being equal (perfectly balanced coin, zero air resistance, etc.) each coin toss has a 50-50 chance of being heads or tails and is in NO WAY dependant on the previous coin toss.

4

Back when my daughter, Casey, was 6 or so, she and her friend argued about who would get the front seat in the car. I said we’d flip for it. She said, “Noooooooooooo, I always lose. Let’s guess rocks instead.” So, to show her that she probably just had stronger memories of the times she’d lost, I told her we’d toss the quarter twenty times and write down how many times she ‘called it’ correctly.
She managed to call it wrong 21 times in a row. By the 12th try, she was very upset, but I begged her to persist so we could see how big of a wrong streak she could build up. By 20, she was crying quietly. I wanted her to go on-- she complied once more, started bawling, and I didn’t have the heart to ask her again.
We chose rocks, and Casey won. (Yes, we had to have a playoff, since Naomi now was convinced that she would have won had we gone ahead with the coin toss. I forget who won the playoff.)
-Sue

5

The coin doesn’t remember. Even if you’ve tossed a fair coin 99 times and by some fluke gotten 99 heads, the chance that the next flip will yield a heads is still one half.

The confusion comes in because you’re comparing statements about two different time periods. The statement “it’s very improbable that you’ll get 100 heads in a row” applys before you’ve begun flipping the coin.

However, the statement “if I’ve gotten 999 heads in a row, the chance of my next flip giving a heads is .5” is a statement about a later, different time period, a time when you’ve already gotten 99% of the way to a string of 100 heads.

6

No, that’s not really how it works. Think of it this way:

Say I offer to play a game with you–we’ll take a coin and flip it over and over again for, say, 1000 times. After each flip, we’ll count the total number of heads and the total number of tails we’ve had throughout the game. If there are more heads, you pay me a dollar; more tails, I pay you a dollar. If we have the same amount of heads and tails at that point, neither pays the other, and we just go on to the next round.

Now obviously, if we use a fair coin, this is a fair game. But what do you think will be the most likely scenario? Will we end up just passing dollars back and forth, each of us breaking about even in the long run? Or will one of us shoot far ahead, winning just about everytime?

It turns out to be more likely than not that one of us will shoot out far ahead. It’s likely that there will be “streaks” of heads (or tails) that will put one side ahead of the other for a considerable while (if not the duration of the game, in fact).

Think of it this way. Suppose early on tails gets ten points ahead of heads. That means you’re going to be winning for quite a while (at least the next ten rounds, probably more). In the long run (and by that I don’t mean simply the next flip), the ratios of heads to tails will approach 1:1. Say that 10 point lead sticks around for the entire game–then for the entire game, we’ll end up having 505 tails, 495 heads–pretty close to 1:1.

What I’m trying to say is that streaks will occur, but their significance vanishes in the long run. If heads gets ahead by 50 points, there will be somewhat of a tendency for that 50 point lead to remain for a good while. However, as we flip more and more, that 50 point lead will be insignificant as far as the ratio of heads to tails goes, which will tend to 1:1.

Does that answer your question?

7

The question really has been answered but the two things people often confuse about stats is

  1. You look at the same data but ask two different questions

  2. You don’t have to get the probablity.

Let’s look

Number 1. Let’s say you throw a coin and 19 times in a row it comes up heads. What is the probablity it will come up heads on the next toss. 50%.

But what you are really asking yourself is what is the probablity after 19 heads in a row it will come up heads again. That is an entirely different question. It is like asking yourself what are the odds I can flip a coin and have it come up 20 times in a row as heads.

See where the confusion lies.

Let’s take number two.

You may NEVER get the probability.

There is nothing in the laws of statistics that says even in a fair coin toss it won’t always come up heads. The odds are it WON’T but it still could.

See how it’s confuses people it’s saying it should, and it would, but it doesn’t HAVE to.

8

[random bit of trivia]
If you flip a penny 10000 times, it will only be heads about 4950 times, on average; the heads picture weighs more and will hence be more likely to be on bottom. So always guess heads!
[/random bit of trivia]

9

God explanations here…

http://www.csicop.org/si/9809/coincidence.html

http://www.csicop.org/si/9809/coincidence.html

10

The strict math interpretation of the problem is this:

We take a random variable

V[sub]N[/sub]

equal to the difference of heads minus tails so far in N tosses of the fair coin. As N grows to large values the value of

V[sub]N[/sub]

approaches the square root of N. Thus, as Cabbage pointed out, in a single game one person will tend to get ahead and stay ahead after a suitably large number of tosses.

But, the average value ofV[sub]N[/sub], defined by

V[sub]N[/sub]/N

approaches zero, since

V[sub]N[/sub]/N = 1/N[sup]1/2[/sup].

Thus the average value of the heads/tails ratio approaches unity even as the absloute value diverges from a zero difference.

11

personally I think , that there is really really no such thing as 60% chance that this will happen or 40 % chance that this will not happen. All probablities are just 50% , either you will pick a red ball out of 100 blue balls (without seeing ) or won’t. Why so much fuss about probablity

12

Afghan - I would like to play poker with you some day

13

Afgan, I think you misunderstand probability. Just because there are two possible outcomes does not mean that the outcomes are equally likely. Take you example. Let’s say you had a second bag with one blue ball and 100 reds. Do you think that over the long term you will pick as many red balls out of the first bag as you will pick out of the second?

14

You’re joking, right?

15

Patient: So doctor, I understand that this medicine has a very good chance of curing me, so it’s basically 50-50 right?
Doctor: No, not at all, this medicine is very effective against your condition, the chances of a complete cure are more than 99.99%; complications or failures, are uncommon in the extreme, especially for someone of your age and general condition.
Patient: But they do happen?
Doctor: Well yes, there is always the slim chance of failure but I have to say that in the case of this particular tratment that chance is vanishingly small; this is a very common procedure and we haven’t seen a problem with it in this hospital for more than twenty years.
Patient: So it could go either way, right?
Doctor: Certainly I’d be lying to you if I said there was no possibility of any complications, but as I said, it’s very very improbable indeed, furthermore, since we have caught the disease in it’s early stages, the probability of a full and speedy cure is compounded; I’m very confident that this will help you
Patient: Ah, I understand now…
Doctor: Good, so I’ll prescribe…
Patient: (interrupting) Either I will get better, or I won’t, it’s 50-50
Doctor: (removes gun from desk drawer and fires it into patient’s twitching body until the clip is empty)

16

I was never that good at stats and I am confused, but I think we are talking about 2 different probability models.

The single coin toss probability is (random trivia aside), 50/50: “X is a discrete random variable that can have two values, and occurrences of the events X counts are independent. The probability distribution of X is a binomial distribution”

For a series of throws over a period of time, I think we’re talking Poission probability distribution: “a probability distribution showing the probability of X occurrences of an event over a specified interval of time or space.”

Hoping it will be prophetic, I attach my signature…

17

…bizarre. After preview, the tick disappears…

Trying again…

18

Wow. Do you realize that the chances of doing that are 1 in two million? I suggest you contact James Randi and see if your daughter can win his million dollar prize.

Um, by the way, if your daughter had to pick five numbers from one to fifty, and then a sixth from one to thirty-six, which numbers would she not ever choose? I’m just, y’know, curious.

19

They’re both binomial distributions. The first one has parameters 1 and .5, and the second one has parameters n and .5.

20

Dear zut alors,
Hmm, maybe I’m the one with super powers. Maybe I was reading her mind to see what she’d ‘call’ next, then flipping the coin just so, controlling which side landed up. Yup, that’s it. So, I’m going to apply for the prize! Naah. Just a bizarre coincidence. Even if it was 1 in 2 million – I mean, people do win the lottery, and it doesn’t mean they’re super-powered-number-pickers.
I don’t understand how it would be possible to answer your number question. How would you ever know which numbers you would not ever choose, unless you did this thing a mess of times and kept track? Even then, you’d never know if any of the numbers that you’d not chosen yet would be numbers you’d NEVER choose again. Or were you just joshing me?
Sue
p.s. Whoops! That was close! I “get it” now. I was previewing, and saw that my reply contained the clue I needed (hey, it’s early). You’re really funny! How about this – I’ll ask her which ones she would choose, and you can ask the lottery for their reciprocals. Don’t take “no” for an answer.