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.

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 or similar polarization of outcomes in the sample data which places them above average odds.