Question

A surgical technique is performed on seven patients. You are told there is a 70% chance of success. Use a table to find the probability the surgery is successful for exactly five patients.

Answer 1

31.77% probability the surgery is successful for exactly five patients.

Explanation:

For each patient, there are only two possible outcomes. Either the surgery is successful, or it is not. The probability of the surgery being successful for a patient is independent of other patients. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

A surgical technique is performed on seven patients.

This means that
`n = 7`

You are told there is a 70% chance of success.

This means that
`p = 0.7`

Find the probability the surgery is successful for exactly five patients.

This is P(X = 5).

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 5) = C_(7,5).(0.7)^(5).(0.3)^(2) = 0.3177`

31.77% probability the surgery is successful for exactly five patients.