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.