In a shuffled deck of 52 cards, we remove 1 card (we don’t know which one) and use LINEAR SEARCH to find the two of clubs.
a. What is the average cost to find the two of clubs, if it is guaranteed to be in the deck?
b. What is the average cost if the two of clubs is equally likely to be in the deck, or not in the
deck?
c. What is the average cost if it is equally likely that the 2 of clubs is not in the deck or that it is
in the deck?
d. What is the average cost if the card that was removed was the 2 of clubs?
To answer these questions, we need to understand how linear search works and consider the given scenarios.
Linear search is a simple algorithm that checks each element in a collection, one by one, until the target element is found or the search ends. It is commonly used when the collection is not sorted or when the search target's position is unknown.
In the given scenarios, we have a shuffled deck of 52 cards, and we need to find the two of clubs using linear search. Let's analyze each scenario:
a. Average cost to find the two of clubs, if it is guaranteed to be in the deck:
In this case, since the two of clubs is guaranteed to be in the deck, the linear search will always find it. The maximum number of comparisons needed in a worst-case scenario is 52 (when the card is in the last position). Hence, the average cost would be (1 + 2 + 3 + ... + 52) / 52, which equals 26.5 comparisons on average.
b. Average cost if the two of clubs is equally likely to be in the deck or not:
In this scenario, there is a 1/52 probability that the removed card is the two of clubs, and a 51/52 probability that it is still in the deck. If the card is in the deck, the average cost remains the same as in scenario (a) since it is guaranteed to be there. If the card is not in the deck, the search will run through all 52 cards without finding it. Therefore, the average cost would be (51/52) * 26.5 + (1/52) * 52 = 26.038.
c. Average cost if it is equally likely that the two of clubs is not in the deck or that it is in the deck:
In this scenario, there is a 1/53 probability that the two of clubs is not in the deck, and a 52/53 probability that it is in the deck. If the card is in the deck, the average cost remains the same as in scenario (a). If the card is not in the deck, the search will run through all 52 cards without finding it. Therefore, the average cost would be (52/53) * 26.5 + (1/53) * 52 = 26.075.
d. Average cost if the card that was removed was the two of clubs:
Since we already know that the two of clubs was removed, the search will run through all 52 cards without finding it. Hence, the average cost would be 52 comparisons.
By considering these scenarios, we can determine the average cost to find the two of clubs using linear search in each case.