The probability subject is a very difficult subject to me. This is because it involves estimation of all the possible events. Therefore, it involves the combination and permutation. And there is no exact formula for different situations. It also involves statistics.

Gambler’s fallacy, is a very good notion. To simplify it, gambler’s fallacy is a belief that the next outcome will be different if the observed outcome is repeated consecutively, where these events are actually independent. The best example is tossing the coin, which has the probability of 0.5 for head and 0.5 for tail. Because tossing the coin first time will not affect the second time, the probability to get the head or tail is always same.

For example, first tossing the coin to get the head is 0.5, then 2nd for head is 0.5*0.5 = 0.25, then 3rd for head is 0.125. As for the gambler’s fallacy, the person will think that the probability to get another head is 0.0625, which the chance is very small. Thus, the person will assume that the next one is tail. However, in the actual sense, because of the events are independent, thus, to get the 4th time as tail, it is also 0.25 * 0.5 = 0.0625. That means, whenever we toss the coin, the probability to get head or tail is always 0.5.

However, recently I think about the probability again in the empirical way. Firstly, we need to know, the probability 0.5 means that, if we toss the coin 1000 times, the result of head is approximately 500 times. The greater the number of tossing, the results will be more close to 0.5. However, if the total number of tossing decreases, the deviation of the empirical result becomes higher. For example, if we toss the coins only 2 times, we might get 2 tails for both tossing, where the empirical result of head is 0.

So, that is why gambler’s false assumption happened. If a person is going to toss the coin 500 times, and this results 250 tails successively, that means the next 250 toss must be heads, so that the empirical result will be 0.5. This is interesting part. This kind of belief normally connected to the fate or luck. That is why some people believe that if we are too lucky successively, we might finish using our good luck for our whole life, then we will left only bad luck until the end of the day.

As a conclusion, the **gambler’s fallacy is true** (refers to 2nd and 3rd paragraphs). But sometimes we cannot accept it, for example tossing the coins and get the head 10 times successively, then the next 10 toss are probably tails, so that the probability will equal to 0.5, this is what we normally believe. Therefore, if asking me to guess the next outcome after 10 successive heads, I will also guess tail, even I know that gambler’s fallacy is true.

There are a couple of important points. Firstly, probabilities do not tell you what is going to happen, they merely tell you what is likely to happen. It is unlikely that you will toss twenty coins and that they will all come up heads. But if enough people toss enough coins for long enough, then this may well happen. It will startle the person it happens to, but think of all the people it didn’t happen to! Secondly, if you toss a coin nineteen times and it comes up heads each time, then it is not more likely that the next toss will be a tail. The odds stay the same, at 50%. The tosses are called ‘independent events’ which means that the coin can’t remember what has happened to it. While twenty heads in a row is unlikely, once you have nineteen heads in a row, the unlikely event has already happened. The potential twentieth head has the same probability as the first head. Another way of looking at it is that any sequence of twenty tosses is unlikely as twenty heads in a row, even if it looks random. But you have to write down the sequence before you start tossing to see if you get it!

until a respectable analysis of probability is forthcoming;it is unclear that the gamblers fallacy is a fallacy (in the infinite long run). Technically Von Mises/reichenbachian/maximum entropy style frequentist interpretations are committed to a teleological kind of gamblers fallacy; despite their best attempts at defining randomness, and place selection rules/free from after effect rules. Any analysis committed to the fallacy of the gambler fallacy is non reductive and those which are reductive are not informative in the finite case, and are committed to it. Prob 1 does not mean of ‘of necessity’

I disagree…..yes the probability is 50-50 but the degree of certainty has changed…..as the streak continues the end is closer…..

“Therefore, if asking me to guess the next outcome after 10 successive heads, I will also guess tail, even I know that gambler’s fallacy is true.”

Then you will be wrong about half the time in such guesses, which is as good as you can hope for with any other strategy as well. One reason the gambler’s fallacy persists is that in many cases it leads it doesn’t make out predictions any worse than chance.

Correct. I am wrong because that is gambler’s fallacy. The best solution to play in gambling is not luck, but statistics.