# hey, i'd really appreciate some help with this question:

A medical treatment is successful 75% of the time. A hospital wishes to treat a group of patients. How many patients should they take so that the probablity that all are cured is 1/10?

I know that p=0.75 and q=0.25 but i think i need to find n next to solve it but i'm not sure how i can find what n is equal to...

## Yes, you will need to find n.

Since P(x) = 0.1 and x = n, you can solve for n using logarithms and the binomial probability function:

P(x) = (nCx)(p^x)[q^(n-x)]

Using what is known:

0.1 = (1)(0.75^x)(0.25^0)

Using logarithms, solve for n:

log(0.1) = n log(0.75)

log(0.1)/log(0.75) = n

8 = n

To check:

P(8) = (8C8)(0.75^8)(0.25^0)

P(8) = 0.1

I hope this will help.

## To solve this problem, you can use the concept of binomial probability. The binomial probability formula is given by:

P(X = k) = nCk * p^k * q^(n-k)

Where:

- P(X = k) is the probability of having k successes out of n trials.

- nCk is the combination formula, which calculates the number of ways to choose k items from a set of n items.

- p is the probability of success in a single trial.

- q is the probability of failure in a single trial.

In this case, you want to find the number of patients (n) required to ensure the probability that all of them are cured is 1/10.

Let's set up the equation using the information given:

P(X = n) = (nCn) * (0.75^n) * (0.25^(n-n))

Since you want P(X = n) to be 1/10, we can set up the equation:

(nCn) * (0.75^n) * (0.25^(n-n)) = 1/10

Simplifying further:

0.75^n * 0.25^0 = 1/10

0.75^n = 1/10

To solve for n, we can take the logarithm of both sides:

log(0.75^n) = log(1/10)

Using logarithm properties, we can rewrite the equation as:

n * log(0.75) = log(1/10)

Finally, we can solve for n by dividing both sides of the equation by log(0.75):

n = log(1/10) / log(0.75)

Using a calculator, you can compute the value of n.