Albino rats used to study the hormonal regulation of metabolic pathway are injected with a drug that inhibits body synthesis of protein. Usually, 4 out of 20 rats die from the drug before the experiment is over.

Question

a) If 10 animals are treated with the drug, what is the expected value and
standard deviation of the number of rats that will die from the drug before
the experiment is over.

b) If 10 animals are treated with the drug, what is the probability that at
least 8 will die from the drug before the experiment is over.

c) Suppose that there is an alternative drug that can be used and that about 1 out 1000 rats die from this drug before the experiment is over. If 1500 animals are treated with this drug, what is the probability that less than 4 rats will die from this drug before the experiment is over.

d) Consider experiments with drug from part c), what is the expected number of experiments required to observe the first experiment with a rat that
dies due to the use of the drug before the experiment is over?

If 4 outof 20 die, then the probability of a rat dying (using this relative frequency) is p = 4/20 = .2
Part a) is a binomial distribution. The expected number that would die is
mu = np where n is the number of events, which is 10 here. The variance is given by s²=np(1-p). Take the square root to find the standard deviation.
For b) the probability
p(X=k)= n choose k * p^k * (1-p)^n-k
Calculate that for k=8,9,10 and add those together.
Part c) is similar except now p=.001 and you'll need to recalculat the variance. To find the prob p(X<4) you need to find p(X=0,1,2,3) and add them together.
For d) since p = 1/1000 then 1/p is the expected number of observations before a death is observed.
Be sure to check you text on this. These problems pertain to the binomial distribution which is a fairly important one is prob/stats.
I hope this helps.

Answer 1

a) The expected value and standard deviation of the number of rats that will die from the drug before the experiment is over when 10 animals are treated with the drug is $E=10*\frac{4}{20}=2$ and $SD=\sqrt{10*\frac{4}{20}*\frac{16}{20}}=0.8$.

b) The probability that at least 8 rats will die from the drug before the experiment is over when 10 animals are treated with the drug is $\binom{10}{8}(\frac{4}{20})^8(\frac{16}{20})^2=0.0016$.

c) The probability that less than 4 rats will die from the drug before the experiment is over when 1500 animals are treated with the alternative drug is $P(X<4)=1-P(X\ge4)=1-\binom{1500}{4}(\frac{1}{1000})^4(\frac{999}{1000})^{1496} \approx 1$.

d) The expected number of experiments required to observe the first experiment with a rat that dies due to the use of the drug before the experiment is over is $E=\frac{1}{P(X=1)}=\frac{1}{\binom{1500}{1}(\frac{1}{1000})^1(\frac{999}{1000})^{1499}} \approx 1000$.