The science of uncertainty

I do not want to forget this concept so I try to write this post to share with everyone. Have you thought about the next event when you observe the outcome of BIG or SMALL game in a casino?
In chinese, it is called the game of 大細 or 大小. It is based on three dices total number outcome for each event:
threedicesIf the dice set total number is between 11 and 17, it is considered as BIG and otherwise if the total number is between 4 to 10, it is considered as SMALL. The reason why 3 (1,1,1) is excluded and 18(6,6,6) is excluded because these pattern is consider 圍骰(Alls). So actually the patterns 1,1,1; 2,2,2; 3,3,3; 4,4,4; 5,5,5; 6,6,6 are all excluded from the winning as BIG or SMALL. But we ignore this for the time being. Make it simple.

With the mentioned assumption, we all know that the outcome of any event is either BIG or SMALL, like tossing a coin, HEAD or TAIL.  We can calculate the probability of having HEAD is 1/2 and the same for the TAIL is 1/2. Since the total number of outcome of any event is 2, HEAD or TAIL, BIG or SMALL and when either one happened, the probability is 1/2. That is what most people think.

Here is a situation for people to think what the next outcome can be:

Asumming the previous outcomes are like the following using HEAD or TAIL equivalent with H for HEAD and T for TAIL:


What will the probability of having a H? would it still be 1/2?

From the sample size of 10 samples, 3 are T and 7 are H, statistically the probability of T is 0.3 and probability of H is 0.7 and seems like the coin is biased.  You could guess the outcome for H is 0.7 which is a relatively probable outcome.

Another way to think of that when we try to have another H after 6Hs, the probability of getting a H is 1/2 and having 6 consecutive Hs, the probability is then 1/2 times 1/2 times 1/2 times 1/2 times 1/2 times 1/2 which is 1/26=1/64=0.015625 which is very unlikely.

In summary, which one is probable?
1. 0.5 of getting H next
2. 0.7 of getting H next
3. 0.015625 of getting H next

The science just cannot tell precisely the probability of getting H after 6 Hs has happened. Is the sample-size of 10 is good enough to justify the above logic or not?