Question

In Example 5.3.2, compute the probability that all 10 successful patients appear in the subsample of size 11 from the Placebo group.

Short answer

\(8.39 \times {10^{ - 8}}.\)

Step-by-step solution

Given information

The given example is:

Consider the patients in the clinical trial

whose results are tabulated in Table 2.1. We might need to re-examine a subset of the patients in the placebo group. Suppose that we need to sample 11 distinct patients from the 34 patients in that group.

What is the distribution of the number of successes (no relapse) that we obtain in the sub sample?

Let X stand for the number of successes in the sub sample. Table 2.1 indicates that there are 10 successes and 24 failures in the placebo group. According to the definition of the hyper geometric distribution, X has the hypergeometric distribution with parameters A = 10, B = 24, and n = 11. In particular, the possible values of X are the integers from 0 to 10. Even though we sample 11 patients, we cannot observe 11 successes, since only 10 successes are available.

Define the pdf

Here, as noted, X follows a hypergeometric distribution \(X \sim Hyp\left( {A = 10,B = 24,n = 11} \right)\)

Therefore, the pdf is,

\(p\left( x \right) = \frac{{{}^{10}{C_x}{}^{24}{C_{11 - x}}}}{{{}^{34}{C_{11}}}},x = 0,1,2, \ldots ,10\)

Calculate the desired probability value

The probability that all 10 successful patients appear in the subsample of size 11 from the Placebo group is;

\(\begin{array}{c}p\left( x \right) = \frac{{{}^{10}{C_{10}}{}^{24}{C_{11 - 10}}}}{{{}^{34}{C_{11}}}},x = 0,1,2, \ldots ,10\\ = \frac{{{}^{10}{C_{10}}{}^{24}{C_1}}}{{{}^{34}{C_{11}}}},x = 0,1,2, \ldots ,10\\ = 8.39 \times {10^{ - 8}}.\end{array}\)

Therefore, the final answer is \(8.39 \times {10^{ - 8}}\).