Q.4.26 - A First Course in Probability

Question

One of the numbers $1$ through $10$ is randomly chosen. You are to try to guess the number chosen by asking questions with “yes-no” answers. Compute the expected number of questions you will need to ask in each of the following two cases:

(a) Your ith question is to be “Is it i?” i = $1,$ $2,$ $3,$ $4,$ $5,$ $6,$ $7,$ $8,$ $9,$ $10$. (b) With each question, you try to eliminate one-half of the remaining numbers, as nearly as possible.

Step-by-Step Solution

Answer

1. Your $i$ th question is to be $\frac{11}{2}$
2. The number of question is $3.46$$3.46$

Step 1: Given Information (Part a)

Given in the question that, randomly chosen numbers 1 through 10.

We need to compute the expected number of questions to ask $i$ th question is to be " is it $i$?".

Step 2 : Explanation (Part a)

$E\left[X\right]=1\times \frac{1}{10}+2\times \frac{1}{10}+3\times \frac{1}{10}+4\times \frac{1}{10}+5\times \frac{1}{10}+6\times \frac{1}{10}+7\times \frac{1}{10}+8\times \frac{1}{10}+9\times \frac{1}{10}+10\times \frac{1}{10}$

Because $P$ of every digit $=\frac{1}{10}$

$E\left[X\right]=\frac{1}{10}[1+2+3+4+5+6+7+8+9+10]$

$=\frac{1}{10}\times \frac{10\times 11}{2}$

$=\frac{11}{2}$

Step 3: Final Answer (Part a)

The expected number of question is $\frac{11}{2}$.

Step 4: Given Information (Part b)

Given in the question that, randomly chosen numbers 1 through 10.

We need to compute the expected number of questions with each question, you try to eliminate one half of the remaining numbers, as nearly as possible

Step 5: Explanation (Part a)

We need to apply Binary search here,

$2^{\mathrm{n}}=11$

$n$ is number of questions

$\mathrm{n} \ln 2=\ln 11$

$\mathrm{n}=\frac{\ln {\displaystyle }11}{\ln {\displaystyle }2}$

$=3.46$

Step 6: Final Answer (Part a)

The number of questions is 3.46