The normal person thinks that because the last 20 people survived, the next patient is very likely to die.

The mathematician considers that the probability of success for each surgery is independent, so in the mathematician’s eyes the next patient has a 50% chance of survival.

The scientist thinks that the statistic is probably gathered across a large number of different hospitals. They see that this particular surgeon has an unusually high success rate, so they conclude that their own surgery has a >50% chance of success.

Or do think there’s something special about the person that makes them flip tails more often?

Yes, that’s the conclusion that the scientist has come to. The chance of getting 20 in a row is so extraordinarily unlikely that it’s reasonable to conclude that the chance is not 50/50 for that particular surgeon.

Mathematician sees each individual outcome as independent 50% chance.

Scientist realises that the distribution of failures and successes puts him in a favorable position. e.g. for the 20 in a row to be a success in a 50% fail rate that means the previous 20 were all failures or some similar circumstances where the success rate rose over time.

Can somebody explain the difference between the mathematician and the scientist parts of this?

The normal person thinks that because the last 20 people survived, the next patient is very likely to die.

The mathematician considers that the probability of success for each surgery is independent, so in the mathematician’s eyes the next patient has a 50% chance of survival.

The scientist thinks that the statistic is probably gathered across a large number of different hospitals. They see that this particular surgeon has an unusually high success rate, so they conclude that their own surgery has a >50% chance of success.

Thanks. I suspect a mathematician would consider the latter point too though.

Removed by modAnd me mate paul

My aunt Nancy would surely be in the 50% who die

Would be or do you want her to be? Sound like aunt Nancy isn’t very nice.

The popularity of casinos and lotteries say otherwise.

Someone who goes to casinos would come to the same conclusion, thinking that the surgeon is ‘running hot’.

Anyone with a good high school education should understand this.

deleted by creatorYes, that’s the conclusion that the scientist has come to. The chance of getting 20 in a row is so extraordinarily unlikely that it’s reasonable to conclude that the chance is not 50/50 for that particular surgeon.

deleted by creatorMathematician sees each individual outcome as independent 50% chance.

Scientist realises that the distribution of failures and successes puts him in a favorable position. e.g. for the 20 in a row to be a success in a 50% fail rate that means the previous 20 were all failures or some similar circumstances where the success rate rose over time.