Problem $3(10$ Points$)$ Consider a variant of the Monte Hall game: there are $4$ closed

doors in front of you. There is a gift behind one of the doors equally likely. The game

goes like this: You pick a door first, then the host will open a door which does not have

the gift, which is referred to as the $$empty door$($of course, the host will not open the

door that you pick!$).$ Afterwards, you will be given an opportunity to make another pick

$($you may also choose to stick to the initial pick$).$ What is the chance of winning this game

for each of the following two strategies?

$$ Strategy I: After the host reveals one empty door, do not change your choice;

$$ Strategy II: After the host reveals one empty door, change your choice, and ran$-$

domly pick one of the remaining doors