Chapter 4: Q.4.26 (page 165)

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.

Short Answer

Expert verified

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

Step by step solution

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 $i$ ?".

Explanation (Part a)

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

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

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

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

$= \frac{11}{2}$

Final Answer (Part a)

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

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

Explanation (Part a)

We need to apply Binary search here,

$2^{n} = 11$

$n$ is number of questions

$n \ln 2 = \ln 11$

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

$= 3.46$

Final Answer (Part a)

The number of questions is 3.46

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Short Answer

Step by step solution

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Final Answer (Part a)

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