The average Number of key comparisons done... | গণিত Multiple Choice Question

<html> <head> <title>Sequential Search Key Comparisons</title> </head> <body> <h1>The Average Number of Key Comparisons in a Sequential Search</h1> <h2>Question:</h2>

The average number of key comparisons done in a successful sequential search in a list of length n is:

Choices:

log n
(n + 1)/2
(n - 1)/2
n/2

<h2>Correct Answer:</h2>

(n + 1)/2

<h2>Explanation:</h2>

In a sequential search, also known as linear search, each element in the list is checked one-by-one from the beginning to find the target value. Let's break down why the correct choice is (n + 1)/2:

<h3>Understanding Sequential Search</h3>

Given a list of length n, the best-case scenario occurs when the desired element is the first one in the list, making it only one comparison. The worst-case scenario occurs when the desired element is the last one in the list, making the number of comparisons equal to n.

<h3>Calculating the Average Case</h3>

The average number of comparisons in a successful search can be calculated as the mean of all possible numbers of comparisons:

The position of the target element could be anywhere from the first place (1st) to the last place (n-th). Therefore, the average number of comparisons for a list of length n can be calculated as follows:

The sum of the first n natural numbers is:

$ \sum_{i=1}^{n} i = \frac{n(n + 1)}{2} $

The average number of comparisons is then:

$ \text{Average} = \frac{\sum_{i=1}^{n} i}{n} = \frac{\frac{n(n + 1)}{2}}{n} = \frac{n + 1}{2} $

Thus, the average number of key comparisons in a successful sequential search in a list of length n is (n + 1)/2.

This can be elaborated with a small example:

<h3>Example:</h3>

Consider a list of 3 elements: [a, b, c]. If you search for each element:

Finding 'a' takes 1 comparison
Finding 'b' takes 2 comparisons
Finding 'c' takes 3 comparisons

The average number of comparisons is:

$ \frac{1+2+3}{3} = \frac{6}{3} = 2 $

Which is consistent with the formula:

$ \frac{n+1}{2} = \frac{3+1}{2} = 2 $

<h3>Conclusion:</h3>

Therefore, for a list of length n, the correct answer is indeed (n + 1)/2, as this represents the average number of comparisons performed in a successful sequential search, considering all possible positions of the target element.

For more in-depth study on this topic, consider referring to algorithms textbooks such as "Introduction to Algorithms" by Cormen, Leiserson, Rivest, and Stein (CLRS).

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