|The switch logic only applies when information is added about the set originally not chosen.|
Initial guess is 1:3, meaning other set has winner 2:3. If information is added about other set, eliminating all but one of the choices, then the remaining single choice will be winner 2:3 times.
For the 1:50 and 49:50 example to payoff at 49:50, information would need to be added about 48 of the 49 choices. All but one box would need to be eliminated. (Deal or no Deal...)
It is clear that information needs to be added about the other choices for the switch to payoff. Otherwise, everyone would do this:
Go to buy a lottery ticket, and pick a non repeating set of two digit numbers. Your chances of winning are astronomically low, against astronomically high that the winning number is in the set not chosen. Change your mind, and pick a different non repeating set of two digit numbers. Your chances of winning are not now 'astronomically high.' Without information about the other set, there is no advantage to switching.
Go back to the 3 choice problem. It is true that the unchosen set has a 2:3 chance of having the Gold, but when you second chance pick before the additional information, you only have a 1:2 chance of picking from that 2:3 set. Without the added information before your second chance pick, the arbitrary second choice has the same 1:3 chance as the original 1:3 choice. (Otherwise, all but one of us would win the lottery every day, simply by changing our mind when we picked a lottery number.)
With the additional information, your second chance pick is the better choice.
Even with the 1:50 example, your second choice is marginally better if even just one box is eliminated from the 49:50.
Your chances of the 1st pick are 1:50.
Your chnaces of the second pick are 1/48 x 49/50
or 49/48 of 1/50
If you eliminate 48 of the 49 before making your second puck, your seconf pick is
1/1 x 49/50
Just as, if you eliminate 1 of 2 before making your second pick of 3, your second pick is
1/1 x 2/3
(Edited by Fred Bartlett on 4/08, 5:19am)