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 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 onehalf 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 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 onehalf of the remaining numbers, as nearly as possible.

Key Concepts

Expected Value

This concept refers to the average outcome of a random process if it were repeated many times. In the context of the problem, it is used to calculate the average number of questions needed by taking into account the probability of each number being chosen and the corresponding number of questions required to guess it.

Sequential Search Strategy

This strategy involves checking each possibility one by one until the target is found. Here, it corresponds to asking questions in a fixed order (e.g., 'Is it 1?', 'Is it 2?', etc.), leading to an expected number of questions that depends on the position of the target in the sequence.

Binary Search Strategy

This algorithmic approach involves dividing the search space into roughly equal halves with each question, efficiently narrowing down the possible choices. It minimizes the maximum number of questions needed by strategically eliminating half of the remaining numbers at each step, which is a fundamental idea in algorithm design for searching sorted or partially ordered data.

Decision Tree

A decision tree is a tool for modeling decision processes, where each node represents a test (such as a yes-no question) and each branch represents the outcome of the test. In the context of questioning to guess a number, the decision tree framework helps in visualizing and analyzing the sequence of questions and the corresponding reduction in uncertainty about the chosen number.

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