Adding Up Scenarios

Two Takeaways from This Film · Adding Up Scenarios

	A	B	C
A	AA	AB	AC
B	BA	BB	BC
C	CA	CB	CC

AA

Million in A, guest choses A. 1/9

AB

Million in A, guest choses B. 1/9

AC

Million in A, guest choses C. 1/9

BA

Million in B, guest choses A. 1/9

BB

Million in B, guest choses B. 1/9

BC

Million in B, guest choses C. 1/9

CA

Million in C, guest choses A. 1/9

CB

Million in C, guest choses B. 1/9

CC

Million in C, guest choses C. 1/9

The next choice is of the host. He will not open the door to the million, and he knows which it is, and he cannot open the door the guest proposed, since he wants to ask the guest if he sticks to the choice.

This means, only AA, BB, CC will subdivide.

AAB

Million in A, guest choses A. Host opens B. 1/18

AAC

Million in A, guest choses A. Host opens C. 1/18

ABC

Million in A, guest choses B. Host opens C. 1/9

ACB

Million in A, guest choses C. Host opens B. 1/9

BAC

Million in B, guest choses A. Host opens C. 1/9

BBA

Million in B, guest choses B. Host opens A. 1/18

BBC

Million in B, guest choses B. Host opens C. 1/18

BCA

Million in B, guest choses C. Host opens A. 1/9

CAB

Million in C, guest choses A. Host opens B. 1/9

CBA

Million in C, guest choses B. Host opens A. 1/9

CCA

Million in C, guest choses C. Host opens A. 1/18

CCB

Million in C, guest choses C. Host opens B. 1/18

Now, we need to count together the possibilities that look the same to the guest, who, a priori, does not know where the million is.

AB

Million in A, guest choses A. Host opens B. 1/18

AB

Million in C, guest choses A. Host opens B. 1/9

AC

Million in A, guest choses A. Host opens C. 1/18

AC

Million in B, guest choses A. Host opens C. 1/9

BC

Million in A, guest choses B. Host opens C. 1/9

BC

Million in B, guest choses B. Host opens C. 1/18

CB

Million in A, guest choses C. Host opens B. 1/9

CB

Million in C, guest choses C. Host opens B. 1/18

BA

Million in B, guest choses B. Host opens A. 1/18

BA

Million in C, guest choses B. Host opens A. 1/9

CA

Million in B, guest choses C. Host opens A. 1/9

CA

Million in C, guest choses C. Host opens A. 1/18

For each scenario, the one where the million and the guest chose the same is 1/18. The one where the million is outside the guest's first choice is 1/9, twice as much./HGL