Question

The probability that a patient will recover from a certain blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that at most 5 people will survive?

Answer 1

This is a binomial distribution problem, so lest first identify the probabilty on "winning" and "losing":

It is said that 0.4 recovers, so 0.6 will not recover.

Now we are going to multiply the probabilities using the given information:

`P(x=5)=(15!)/(10!5!)\cdot0.4^5\cdot0.6^(10)`

The first part represents the number of combinations of 5 people in a group of 15, the second one the probability that exactly 5 of them will recover and the last part the probability for those that will not recover.

`P(x=5)\approx0.1859`

This is the probability that exactly 5 will recover, but as the problem says that at most 5 people will survive, we could do the same for 4, 3, 2, 1 and 0.

By doing so and adding the probabilities we get:

`P(x\le5)\approx0.4032`

That is the probability that at most 5 people will survive.