Gambler’s fallacy

Referring to my previous post about gambler’s fallacy, I was totally wrong after I pondering more about this.

In an example of tossing a coin, we know that to get a “tail” is 0.5 probability and “head” is 0.5 probability. That means, each result should fairly appear once. And in the experiment, if we tossed the coin 1000 times, then we will get the result of “tail” appeared around 500 times and “head” another 500 times.

And in my previous post, I mentioned that, if I tossed the coin 10 times, and all the results are “tail”, then, as a gambler’s fallacy, I will feel that next toss or next 10 tosses should be probably “head”, so that the probability will be 0.5 and 0.5.

However, the problem is the “time to start tossing” restricted my thinking, thus I have a feeling as mentioned above.

In the experimental probability, the more we toss the coin, and collect the results, then the more accurate our results. For example, calculating the probability by tossing the coin 1000 times is better than calculating the probability by tossing the coin 100 times. Thus, it is not valid by tossing the coin ONCE and conclude that, “tossing the coin will ALWAYS be head (or tail)”.

Therefore, referring the situation that if I tossed the coin 10 times and all the results are “tail”, it cannot be considered as a reliable data. This is because, “someone” may have tossed the same coin 10,000,000 before me and the the result of probability 0.5 and 0.5. Thus my 10 times and get the “tail” doesn’t mean anything.

Besides that, the experiments are done to get the calculation of the probability, not reversing it by presume a probability and test by the experiments as the situation above. If I am the first person to toss a specific coin 100 times, and all the results are “tail”,  then I can say that the probability of getting the “head” of that specific coin is less than 0.5 and the “tail” is more than 0.5. I cannot simply assume that the next 100 times have the high probability to get “head”. There are several reasons: i) the coin may be poorly designed, it may ALWAYS produce “tail”, and ii) the event of tossing the coin is independent, that is tossing the coin now does not affect tossing the coin next time.

So, my commenter’s statement is very convincing.

Is gambler’s fallacy really a fallacy?

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.

A math problem 6/2(1+2)

Today, I read the newspaper, and found that there were a lot of people discussing this questions. You can read it from Yahoo! Answer. And there are a lot of people answering this expression equal to 9:

6/2(1+2) = 9

I really don’t understand the possibility of answering it as 9. Some of them said based on PEMDAS, it must be 9; also other said based on distributive property, it is 1.

My answer is 1, and it is impossible to be 9. If using variables for the value inside the parenthesis, then we will get

6/2(a+b)
= 6/(2a + 2b)
= 6/(2(1) + 2(2))
= 6/(2 + 4)
= 6/6
= 1

The above is the distributive property.

This is exactly same as 6/-1(3), which should read as 6/(-1(3)), but not (6/-1)(3).