Q. 9.16 - A First Course in Probability

Question

A random variable can take on any of n possible values x1, ... , xn with respective probabilities p(xi), i = 1, ... , n. We shall attempt to determine the value of X by asking a series of questions, each of which can be answered “yes” or “no.” For instance, we may ask “Is X = x1?” or “Is X equal to either x1 or x2 or x3?” and so on. What can you say about the average number of such questions that you will need to ask to determine the value of X?

Step-by-Step Solution

Answer

The average number of questions to determine the value of $X$ is$E\left(X\right)-\frac{n}{2^n}$

Step 1: Given Information

We have to find the average number of questions that you will need to ask to determine the value of $X$.

Step 2: Simplify

 Mark that the answer to a particular question is $\frac{1}{2}$ i.e.

$P\left(\text{ Yes }\right)=P\left(\text{ No }\right)=\frac{1}{2}$

which means$P\left(X=k\right)={\left(\frac{1}{2}\right)}^k$ for $k=1,\dots ,n$. So, the average number of questions is

$E\left(X\right)=\sum _{k=1}^{n}k\cdot {\left(\frac{1}{2}\right)}^k$

To calculate, multiply it by $\frac{1}{2}$ .

$\frac{1}{2}E\left(X\right)=\sum _{k=1}^{n}k\cdot {\left(\frac{1}{2}\right)}^{k+1}$

Subtracting that from the expression in $\left(1\right)$, we have that

$\frac{1}{2}E\left(X\right)=\sum _{k=1}^{n}k\cdot {\left(\frac{1}{2}\right)}^{k+1}-n.{\left(\frac{1}{2}\right)}^{n+1}$

Using the formula $\sum _{k=1}^n{\left(\frac{1}{2}\right)}^k=\frac{1}{2}.\frac{1-0.5^{n+1}}{1-0.5}$ to obtain that the final answer. That is

$E\left(X\right)-\frac{n}{2^n}$