Bidding Theory and Probability of getting highest value in list after ignoring first half of list

Question

You are offering an item for sale and there are n bidders. The bidders appear in sequence. When bidder i appears he or she makes a bid bi > 0. You must decide immediately whether to accept the bid or not. If you accept the bid, the item is sold to the bidder and all other bidders are turned away. If you reject the bid, the bidder departs and the bid is withdrawn. You decide to use the following strategy. You watch the first n/2 bids without accepting any of them. Let b∗ be the highest bid among these. Then, in the final n/2 bids, you accept any bid that is larger than b∗. (If there is no such bid, you accept the final bid). What is the probability that you’ll get the highest bid? Find the answer in terms of n. If n = 50, what is this value?

enter image description here