A benefactor offers you to pick one of 3 closed enevelopes O

A benefactor offers you to pick one of 3 closed enevelopes. One of the enevelopes contains a $100 bill. The other two contain $5 bills. You randomly pick an envelope, but, before you open it, the all-knowing benefactor opens one of the remaining two and reveals it contains a $5 bill. You are then offered the chance to change your original pick. Should you switch your original envelope for the other unopened enevelope?

Solution

Yes,

Let\'s name the envelopes A,B and C for refernce. Consider envelope A as your original pick. Now you have 1/3 chance that it contains $100 bill. There is a 2/3 chance envelope B or C has the $100 bill. If you stick with envelope A, you still have the same 1/3 chance. Now, the benefactor sees one of the envelopes (let\'s say B) which contained $5bill. Your originally taken envelope still has 1/3 chance of having a $100 bill. But by switching to envelope C, you now have a 2/3 chance of getting the $100 bill and your chances will be doubled. So you should switch the envelope.