Explanation:

Merge sort is a popular sorting algorithm that works by dividing an unsorted list into n sub-lists, each containing one element, and then repeatedly merging sub-lists to produce new sorted sub-lists until there is only one sub-list remaining.

The merge operation is the heart of the merge sort algorithm, which takes two sorted sub-lists and merges them into a single sorted list. The number of comparisons performed during the merge operation depends on the length of the two sub-lists being merged.

Let's consider two sorted sub-lists of length 2, say [1, 3] and [2, 4]. The merge operation will compare the first elements of both sub-lists, i.e., 1 and 2, and put the smaller one into the output list. Then it will compare the second elements of both sub-lists, i.e., 3 and 4, and put the smaller one into the output list. Therefore, a total of 2 comparisons are needed to merge two sorted sub-lists of length 2.

Similarly, let's consider two sorted sub-lists of length 4, say [1, 3, 5, 7] and [2, 4, 6, 8]. The merge operation will compare the first elements of both sub-lists, i.e., 1 and 2, and put the smaller one into the output list. Then it will compare the second elements of both sub-lists, i.e., 3 and 4, and put the smaller one into the output list. It will continue this process until all the elements are merged. Therefore, a total of 6 comparisons are needed to merge two sorted sub-lists of length 4.

It can be observed that the number of comparisons needed to merge two sorted sub-lists of length n can be expressed as follows:

T(n) = T(n/2) + T(n/2) + n - 1

Where T(n/2) represents the number of comparisons needed to merge two sorted sub-lists of length n/2, and n-1 represents the number of comparisons needed to merge the two sorted sub-lists into a single sorted list.

Using the above formula, we can calculate the number of comparisons needed to merge two sorted sub-lists of length 2 as follows:

T(2) = T(1) + T(1) + 2 - 1
= 1 + 1 + 1
= 3

Therefore, the average number of comparisons needed to merge two sorted sub-lists of length 2 is:

(T(2) / 2) = 3 / 2 = 1.5

Since we need to merge two sorted sub-lists during each iteration of the merge sort algorithm, the average number of comparisons needed to sort a list of length n using merge sort can be expressed as follows:

T(n) = 2 * T(n/2) + n

Using the above formula, we can calculate the average number of comparisons needed to sort a list of length 2 as follows:

T(2) = 2 * T(1) + 2
= 2 + 2
= 4

Therefore, the average number of comparisons needed to sort a list of length 2 using merge sort is:

(T(2) / 2) = 4 /