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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

Barthélémy Déchanet
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    I would say that your brothers claim is a classic instance of "Gamblers Fallacy" https://en.wikipedia.org/wiki/Gambler%27s_fallacy . You are correct, if the surgeries are independent, then the past has no influence on the present. – Leander Tilsted Kristensen Jan 28 '22 at 13:58
  • I saw this mentioned on r/mathmemes. The way it is presented, it is unclear how the 50% success rate was come to. In reality for something like this the success rate is talking about the number of successful such surgeries performed by all surgeons compared to the number of such surgeries attempted by all surgeons, not this surgeon specifically. The success/failure of one surgery is not independent compared to another as when time advances the surgeons gain more practice and new techniques are developed, the success rate should increase over time. – JMoravitz Jan 28 '22 at 14:00
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    Further, procedures done by the same surgeon are not necessarily going to be independent either. This particular surgeon may well just be more skilled compared to others, the others drive the average down while this surgeon drives the average up. I would absolutely trust that this surgeon will continue to perform successful surgeries as per Bayesian Inference. – JMoravitz Jan 28 '22 at 14:02
  • @JMoravitz To support the assumption of improvements, we would need more context. Runs of length $20$ on a $1/2$-chance do happen (although they are extremely rare). Otherwise, we could easily use this impossibility to win surely in a Casino. – Peter Jan 28 '22 at 14:06
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    https://math.stackexchange.com/q/971913/42969 – Martin R Jan 28 '22 at 14:06
  • @Peter Oh, I wouldn't advocate for using this logic in a casino... absolutely not... but for a procedure that relies almost exclusively on the skill and steady hands of the particular surgeon? One could say that boxer's on the whole win 50% of their matches... Now, was Muhammad Ali just extremely lucky? Or skilled? If Muhammad Ali was pitted against a random boxer, do you really believe it to be 50% chance that he wins? "We could use this to win surely in a Casino..." The payout for betting on Ali wasn't 2 to 1 since everyone expected him to win. – JMoravitz Jan 28 '22 at 14:13
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    The point is that one should be careful about using the independence assumption. In some cases independence of events makes sense such as the flips of a coin. In other cases it certainly does not, as here. – JMoravitz Jan 28 '22 at 14:16
  • @JMoravitz As I said in the initial question, the doctor is right about the fact he have a 50% chance to succeed the intervention. – Barthélémy Déchanet Jan 28 '22 at 14:16
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    @JMoravitz Of course I would not expect a fight in sport to be a $50$% chance in general. But the question indicates that the success chance should be about $50$% considering long term observations. Otherwise, the question is just misleading. – Peter Jan 28 '22 at 14:19
  • @JMoravitz Of course, I refered to Roulette in my example, not to sport bets. – Peter Jan 28 '22 at 14:21

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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.

Sergey Guminov
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