One of the numbers 1 through 10 is randomly chosen. You are...

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 $i$ th 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.

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 $i$ th 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.

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Key Concepts

Expected Value

Expected value is a core concept in probability that represents the average outcome of a random process over a large number of trials. It is calculated by weighting each possible outcome by its probability, giving a single value that summarizes the central tendency of the process.

Sequential Search Strategy

A sequential search strategy involves checking possibilities one by one in a prescribed order. In the context of yes-no questioning, this approach corresponds to asking about each possibility in turn, which results in an expected number of questions that depends directly on the order and uniform probability of each option.

Binary Search Strategy

The binary search strategy is an efficient method for finding a target by repeatedly dividing the set of possibilities into two approximately equal parts. Using yes-no questions to eliminate half of the remaining options at each step leads to a logarithmic reduction in the search space, which reduces the expected number of questions compared to a sequential search.

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