$\textbf{Offer Letter}$
After a whole day of interviews, a HR manager comes with three sealed envelopes. One of them contains an offer letter, and the other two contain rejection letters. You can select one of them and will be hired if you get the offer letter. After you pick one envelope, the HR manager opens one of the other two which contains a rejection letter and offers you a chance to change your mind. Should you change your selection? Explain.
$\textbf{Answer}$
Yes, you should change your mind and pick the other unopened envelop because doing so will double your chance of getting the offer.
$\textbf{Analysis}$
Before the HR manager shows one of the rejection letters, your chance of getting the offer letter is $\frac{1}{3}$.
After the HR manager shows the reject letter, if you do not switch, your chance of luck remains at $\frac{1}{3}$. However, the remaining unopened envelop now has $\frac{2}{3}$ probability of containing the offer letter. This is the combining probability of the two envelops you do not choose initially. As the HR manager eliminates one of these two, the other one will now assume the total probability of these two.
This can also be explained in the following way. Let's indicate the three envelops using $A$, $B$, and $C$. There are three possibilities of which envelope contains the offer letter (the corresponding letters are boxed):$$\begin{array}{rcccc} 1\rightarrow&&\boxed{A} & B & C \\ 2 \rightarrow && A & \boxed{B} & C\\3 \rightarrow && A&B&\boxed{C}\end{array}$$
Without loss of generality, assuming you picked $A$ initially. By not changing your choice, you will only get the offer in the case $1$. However, you will get the offer in both cases $2$ and $3$ if you switch your choice after the HR manager discards one rejection letter.