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.