# Law of Large Numbers, a confusion

According to Law of Large Numbers, if I throw a coin 1000 times approximately 500 will be head and 500 tail. Suppose that I throw the coin 700 times and I got 700 heads. Can I say that in the next 300 throws the probability of getting tails will be higher than probability of getting heads?

Edit: To make an analogy, imagine that you have a box that contains 500 black and 500 white balls. If 700 times (700 exaggerated) you choose black, than it is more probable that for 701 you get white. However if you say me that instead of one box you have 1000 similar box and every time you choose from one box that means the probability of choosing black or white will never change. With independent events you mean this?

Edit2: Imagine that there are billions of people that throwing coins 1000 times. For each person there is an empty box. When he throw the coin, if it is tail he puts a black ball in his box, when the coin is head he puts white ball. So at the end of the experiment there are billions of boxes that each box contains approximately 500 black 500 white ball. So they give me the opportunity to choose one box. The box that I choose represents the one possible coin throwing that I would make. I am asking what is the difference between throwing coins and choosing one box in billions of boxes? If there is no difference, than the first statement holds. For example I pick 400 black balls from my box than it is more probably to choose white ball from remaining 600 hundreds.

• Regarding the Edit: Yes, exactly. – fgp May 24 '13 at 16:54
• The first ball scenario is different. If I remove 500 black balls from a box containing 500 black balls and 500 white balls, then the probability of selecting a white ball is 100%. If you chose a ball, and then put it back however, then the probability would not change. Or as you said in your second ball scenario, if there were 1000 boxes filled with the same amount of ball with the same ratio of colours then the probability would be equal if for every box you took one ball and moved onto the next box to select another ball. – Sujaan Kunalan May 24 '13 at 16:55
• The law of large numbers doesn't tell you that if you throw a coin 1000 times, you well get approximately 500 heads/tails. The latter statement is true but what it tells you is that the more coin tosses you do, the closer the ratio heads/total and tails/total would get to 1/2. More formally, the probability distribution of those 2 numbers will have a decreasing variance and a higher spike. – Youcha May 24 '13 at 17:19
• To argue in a different direction entirely: there are many who would contend that if you flip a coin 700 times and get 700 heads, in your next 300 throws the probability of getting heads will be higher than the odds of getting tails - because the results of your previous 700 flips make it more likely that the coin is biased! Look up Bayesian Inference if you're interested in more information on this perspective. – Steven Stadnicki May 24 '13 at 17:24

## 5 Answers

The way the law of large numbers works isn't by cancelling variations you've already gotten. Instead, it just keeps adding new values until the variations become insignificant.

So if you've gotten 700 heads, every flip still has a 50% chance of heads or tails, which means you'll on average have 700 more total heads than tails for the rest of your flips, no matter how many more you do. The law of large numbers says this is true even if you do 1 million more flips, in which case you'll have about half heads and half tails, since at that point a difference of 700 isn't very much.

Perhaps the most likely explanation for a coin coming up heads 700 times in a row is that it is a two-headed coin. But let's ignore that. Suppose that the coin is fair and that heads and tails both have probability $\frac{1}{2}$ of occuring.

The Law of Large Numbers does not predict that the coin will "make up" for the 700 heads by being more likely to come up tails later.

The Law of Large Numbers does not say here that the gap $$|Number \ of \ Heads - Number \ of \ Tails| \to 0,$$ as you do more and more coin tosses. This is what would happen if the coin could "remember" the past results and correct for them later. This does not happen. Each coin toss is independent and, no matter what happened in the past, the probability of getting tails in the next coin toss is still $\frac{1}{2}$.

Instead it says that $$\frac{Number \ of \ Heads}{Total\ Number \ of \ Coin \ Tosses} \to \frac{1}{2},$$ as we do more and more coin tosses.

To see the difference, suppose we now flip the coin another $2,000,000$ times. Let's guess that there will be $1,000,000$ heads and $1,000,000$ tails (for a total of $1,000,700$ heads and $1,000,000$ tails. Then we have $$\frac{Number \ of \ Heads}{Total \ Number \ of \ Coin \ Tosses}= \frac{1,000,700}{2,000,700} \approx \frac{1}{2}.$$ The gap between heads and tails is still $700$; it did not make up for the initial 700 heads with extra tails later. But we see that final outcome is closer to the probability of $\frac{1}{2}$.

Any two coin flips are independent events, and the outcome of one will not affect the outcome of the other.

In other words, the coin has no memory, and doesn't change its composition or shape from toss to toss, so tosses $701-1000$ will not be affected by the outcomes of tosses $1-700$.

Coin tosses are independent events- the probability will still be 0.5 for tails (assuming fair coin). Keep in mind that if something is an independent event, that any tosses before it won't change its probability of occurring. The Law of Large Numbers pretty much says that in small number of trials you may not get the true/accurate results but if you increase the number of trials, then the results would be more accurate. That being said, if over 1000 tosses, all were heads, it is very likely that the coin is unfair.

Just to add to the other answers given (which I completely agree with), I wanted to mention that this is sometimes referred to as the gambler's fallacy.

And in response to your recent edits: say we have a box that contains 500 black and 500 white balls. Then (if we replace the ball we pick each time so that there are always 500 of each) there will always be an equal probability of picking black or white (assuming the picking process is truly random). Picking 700 black balls in a row doesn't change this probability ( although 700 in a row is indeed astonishingly unlikely to happen and we might then start to wonder if we are really picking randomly or if something, e.g. the way the balls are arranged in the box, is affecting our choice, etc). But if it is indeed random then it is not more probable to get white on pick 701. However, if we do not replace the balls then the more black balls we pick the less likely it becomes to pick another (although we could never pick 700 of them if there are only 500 to begin with). Hope that helps.