Bernoulli Principle

Question

A certain drug was developed, tested, and found to be effective 80% of the time. Find the probability of successfully administering the drug to at least 10 out of 11 patients. (Round your answer to four decimal places.)

Answer 1

I just wanted to add to what Hugh said above.  If using Excel isn't your thing, and you would prefer to calculate by hand (with help of a calculator), here is what the equation looks like:

 

P(X=10 or X=11)=P(X=10)+P(X=11)=[(0.8)¹⁰+(0.2)¹]+[(0.8)¹¹+(0.2)⁰].

 

When you work this out, you get the same answer as Hugh did above, but it maybe be helpful to see it written out like this.

 

Hope this helped!

Brenda

Answer 2

Hi Stephanie,

 

We assume 1) independent outcomes at each trial, 2) constant probability of effectiveness of 0.8. These two assumptions imply that in N trials, the random variable X = number successes follows a binomial distribution. We use Excel to find the answer by using the Excel function BINOM.DIST(number_s, trials, probability_s, cumulative) where the arguments are as follows:

 

number_s = number of successes, call it X for now

trials = number of trials = 11 here

probability_s = probability of success = 0.8 here

cumulative: if you put "TRUE" in here, the function returns the cumulative distribution, which is the probability of getting X or fewer successes in 11 trials. If you put "FALSE" in here, then the function returns the probability of obtaining exactly X successes in 11 trials.

 

There are several different ways to answer this using BINOM.DIST and I will show 2 ways.

 

First, we want the probability of 10 or 11 successes. Therefore, we can just find the probability of 9 or fewer successes and subtract it from 1 as follows:

 

1 - "=BINOM.DIST(9,11,0.8,TRUE)" = 1 - 0.6779 = 0.3221

You can paste the quoted text in the formula bar to obtain 0.6779 as the probability of 9 or fewer successes, and 0.3221 is then the probability that remains for 10 or 11 successes.

 

Alternatively, you can use the BINOM.DIST function and just add the probability of 10 successes and the probability of 11 successes together as follows:

 

"=BINOM.DIST(10,11,0.8,FALSE) + BINOM.DIST(11,11,0.8,FALSE)" = 0.3221

Again, you can just paste the quoted text in the formula bar to obtain the results.

 

Hope this helps. Feel free to ask questions.

 

Kind regards,

Hugh