I read about the law of small numbers (psychological phenomenon related to the statistical law of large numbers), where the following example is given:
Suppose a drug is effective in 80% of patients. If five patients are treated, how many will respond?
Many people reason that 80% means 4 out of 5, so if 5 people are treated, exactly 4 will respond. Always.
Others understand that things are not guaranteed to work out so neatly, but they still believe that it is highly likely that 4 people would respond. Maybe a 90% chance.
In fact, there’s only a 41% chance that exactly 4 would respond out of a sample of 5.
Can someone explain how they got to 41%, by way of example:
If there is a 50% chance of heads and you flip 10 times whats the chance that you get 5 heads?
The number of patients who respond to the treatment follows a Binomial distribution with $n=5$ and $p=0.8$. The probability of exactly four patients responding is $$ {5 \choose 4} \times 0.8^4 \times 0.2 = 0.4096 $$