91Ó°ÊÓ

\({E_1}\) is the event that the first patient is a success.

Consider a clinical trial experiment for the study of treatment for depression.

Consider the decline as a failure and no decline as a success. We consider only patients of the imipramine treatment group. Let us consider pas the proportion of successes among all patients who might receive the treatment.

Let us select the 40 imipramine patients in this trial. Let us consider that there are 11 different possible compositions of the collection of patients.

Let the proportion of success for the 11 possible compositions are:

\(0,\frac{1}{{10}},...,\frac{9}{{10}},1\)

Let partition the sample space by the possible values of p.

Define the event\({B_j}\)represent the event that our sample was drawn from a collection with the proportion\(\frac{{\left( {j - 1} \right)}}{{10}}\)of successes.

Let’s consider the following prior probabilities for the event \({B_1},...,{B_{11}}\) as follows:

Let’s define the event\({E_1}\), representing the first patient is a success. Therefore, by using the prior probabilities of\({B_j}\)and the law of total probability, we compute the probability\({E_1}\)as follows

\(\begin{aligned}{c}\Pr \left( {{E_1}} \right) &= \sum\limits_{i = 1}^{11} {\Pr \left( {{B_j}} \right) \times \frac{{\left( {i - 1} \right)}}{{10}}} \\ &= 0.00\left( {\frac{0}{{10}}} \right) + 0.19\left( {\frac{1}{{10}}} \right) + ...\\ &= 0.356\end{aligned}\)

Now, compute the probability\({E_1}\)on the basis that the probabilities of\({B_j}\)is\(\frac{1}{{11}}\)for each j. In this situation, the probability\({E_1}\)as follows

\(\begin{aligned}{c}\Pr \left( {{E_1}} \right) &= \sum\limits_{i = 1}^{11} {\Pr \left( {{B_j}} \right) \times \frac{{\left( {i - 1} \right)}}{{10}}} \\ &= \sum\limits_{i = 1}^n {\frac{1}{{11}}} \times \frac{{\left( {i - 1} \right)}}{{10}}\\ &= 0.5\end{aligned}\)

It is observed that the probabilities of\({E_1}\)is less for prior probabilities of\({B_j}\)as compared to that for probabilities of\({B_j}\)is\(\frac{1}{{11}}\)for each j. Also, it is observed that prior probabilities are very smaller for the larger values of i.