Shouldn't if you switch or not depend on if you originally chose first or second? If you chose first then the switching probability is the same as the original problem.
P[win car without swapping] = 1/3, P[win car after swapping] = 2/3
If you chose second then you have
P[choose car]= 2/3*1/2 = 1/3 P[choose goat]= 2/3*1/2 = 1/3
There is a 1/3 chance your friend chose the car and a 1/3 chance that you chose the car. So that means before the hosts kicks someone off that there is a 2/3 chance someone has the car.
P[pick car initially]= 1/3
P[win car don't swap/car initially]= 2/3
P[pick car initially]= 1/3
P[win car swap/car initially]= 0
P[pick goat initially]= 1/3
P[win car don't swap/goat initially]= 0
P[pick goat initially]= 1/3
P[win car swap/goat initially]= 1/3
The only way to lose by staying is to originally pick the goat. Otherwise you or your friend have to pick the car and since your friend got kicked off then he can't have the car.
P[win car without swapping]= 2/3, P[win car after swapping]= 1/3 (for choosing second)
Please correct me if I am way off.
