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I know the statement that the previous event won't affect the current event.

But what if the statistic of the previous results(large amount of data) shows that the probability of one particular ball appears is higher than other balls (for example if there are five balls numbered as 1,2,3,4,5. The statistic shows that ball number 5 has a higher probability than 1/5, since the balls are not perfect sphere). Then I can state that I have a higher winning chance if I choose ball number 5.

It seems the previous result do affect my choices. Is there contradiction between those two statements？

2 Answers 2

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If $5$ is actually more likely, than sure, you can. However, just because $5$ has come up more in the past doesn't necessarily mean that it is actually more likely. It could just be that it came up more likely due to random chance. For instance, if you flipped a coin $100$ times and $52$ times it was heads you wouldn't want to conclude the coin was slightly biased towards heads. It could just as easily be a perfectly fair coin; getting $52$ heads out of $100$ is a perfectly likely outcome for a fair coin. However, if it had come up $90$ out of $100$ heads, it would probably be cause to infer that the coin is biased. For cases somewhere in the middle there are methods to quantify what the likely bias of the coin is. This is a broad subject called statistical inference.

That being said, it's possible that there's a bias in the lottery, but I think it's unlikely that it's large enough to be exploitable for large profits or even distinguished from random chance without careful work and a lot of data. Otherwise, people would catch on and they'd have to change it. This is, however, outside the scope of math/statistics.

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The fact that the repetitions of the experiment are independent does not imply that all the events have the same probability. For instance, consider a bag full of balls. There are 4 green balls and 1 purple ball. Each time, you take one ball, put it back in the bag and shake the bag again for the balls to mix. With time, you will obtain a good approximation that the probability of a green ball is 4/5, and the probability of a purple ball is 1/5. However, this does not mean that one repetition has any effect on the next one. When you pick up a ball and put it back in the bag, the probabilities keep being 4/5 and 1/5. So, in your case, if the lottery balls are tricked somehow and the lottery is not fair, you may observe some numbers occur with higher probability than others, but this has nothing to do with independence from one day's lottery to the next day's.