The source of the question below is Edgenuity's Statistics course.
A medical device company knows that the percentage of patients experiencing injection-site reactions with the current needle is 11%. What is the mean of X number of patients seen until an injection-site reaction occurs?
A. 3.1289
B. 8.5763
C. 9.0909
D. 11
Firstly, I would like help in finding the correct answer, not the answer itself. Secondly, the answer appeared to be a no-brainer to me because I am not given the sample size that would be included in the equation $μ = n \cdot p$. I simply converted 11% into the decimal $0.11,$ which, yes, is the proportion, not necessarily the mean. Unless $11$ is the correct answer, I do not know where my errors are. Could you guide me on the beginning steps to the correct answer?
The formula $\mu = np$ only applies if the values of $X$ are observed as a result of a binomial experiment (or to be precise, $X$ is a binomial random variable). For a binomial experiment, all of the following conditions must be satisfied:
The first condition above does not hold, so $\mu = np$ cannot be used.
Instead, $X$ follows a geometric distribution (though, note, there are two types of geometric distributions in practice). I assume you've probably been provided the formula for its mean and can input the appropriate probability.
To elaborate, in the case where $X$ is the number of failures including the first success, the mean is given by $$E(X) = \dfrac{1}{p}\text{.}$$ If $X$ is the number of trials (that is, the number of failures excluding when the success occurs), the mean is given by $$E(X) = \dfrac{1}{p} - 1 = \dfrac{1}{p} - \dfrac{p}{p} = \dfrac{1-p}{p}\text{.}$$ In this case, our "success" is when the injection-site reaction occurs.
As it stands, the question is ambiguous. The way I interpret the question is that the correct answer should be $$E(X) = \dfrac{1-0.11}{0.11} \approx 8.0909$$ because we are interested in the number of patients seen before an injection-site reaction occurs. However, if we include the patients seen before the injection-site reaction occurs and the patient for which the injection-site reaction occurs, the answer would be $$E(X) = \dfrac{1}{0.11} \approx 9.0909$$ yielding answer (C).
If this is the correct answer, the question should really be reworded to say something like "What is the mean of X number of patients seen until and including when an injection-site reaction occurs?"