PLEASE SHOW ALL THE STEPS THX FOR THE HELP Each of the n box

PLEASE SHOW ALL THE STEPS~~~ THX FOR THE HELP!!

Each of the n boxes contains a cheque for a certain sum of money, that can be any positive real number. Boxes are sealed and you can not see the cheques. You open the first box and look at the sum on the cheque. If you like it -you take it. and do not open any more boxes; if you do not like it, then this cheque is destroyed (that is, no coming back), and you move on to the second box. The game continues in a similar fashion until the first time you agree to take the cheque. Ideally, you would like to pick the largest cheque, but it does not necessarily happen due to the fact that you discard a cheque once you decide to move on. Consider the following strategy: for a certain k n - 1 discard the first k cheques - but note their numbers anyway -after that keep opening until the first time you find the box which is better than all the previous ones...or until you reach the last box. For example if n = 9, k = 3, and the initial amounts were then you\'d discard first k = 3 cheques; then the fourth, then the fifth, and will stop on sixth, whose number 46 is greater than all the numbers you\'ve seen before. If, (for the same k and n), the boxes looked like then you\'d discard the first 3 and would keep opening until you reach the very last box, getting only $1... (oops!) So, our question is: for n = 4, find the optimal value of k that, using the above strategy, maximizes the probability of picking the largest (out of n) cheque. Say, if k = n - 1, then you will only pick the last cheque, and the probability for it to be the largest is 1/n. Can we do better? How hard is it to repeat the problem for n = 5?

Solution

The optimal value of k=1 as if we consider a geometric distribution the mode arise at the first point i.e the the first point has the maximum chance of occurence of a success, So considering highest amount of cheque as the success, following the geometric strategy conidering the first box is the optimal decision.