I argued with my brother, a math teacher. He presented a situation where a doctor would perform a procedure on you with a 50% chance of success, but has done it 20 times in the past and has a 100% success rate.
For me he had a 50% chance of succeeding, 50% of failing (considering he was right that there was a 50% chance of succeeding and the last 20 times were luck). But my brother told me that according to the law of large numbers, we had to run away because he would have a good chance of missing this time.
I'm pretty sure I'm right, but he has a degree in math, so I wanted an outside opinion.
Edit: My brother admitted he was wrong, thank you guys :)
New Edit: My question was closed because of clarity. I dont really see what is unclear there and a discussion with someone in the comment section is the only thing that come in my mind. So i repeat it : THERE IS 50% CHANCE FOR THE DOCTOR TO SUCCEED THE INTERVENTION, THE LAST 20 TIMES WAS LUCK, HE IS NOT BETTER THAN THE OTHER DOCTORS, THIS IS MATH NOT THE REAL LIFE
This is a classic incorrect interpretation of the law of large numbers, called the Gambler's Fallacy. The law of large numbers states that the average of a large amount of realizations of a (not too pathological) random variable is likely to be close to the expected value.
In your situation, your brother is basically arguing that if you toss a coin 20 times and get heads every time, then the next time you are more likely to get tails. This is completely and utterly wrong!
If you get heads 20 times out of 20, the only thing the law of large numbers tells you is that that you got pretty (un)lucky and it does not tell you anything at all about what will happen in the future, even probabilistically.