Solution to the Probability Problem

When most people read this question (including me) they tend to assume is very straight forward and has a simple answer. The logic is "Since there are now only two stocks left, one of which is the dog and the other is the ten bagger then there is a 50/50 chance of being a winner. Therefore, staying with your original selection or changing to the other provides the same probability of winning".

This is actually incorrect. In actual fact by having a policy of swapping if given the option then you have a 2/3 chance of winning. To illustrate this simply I'll call the stock C the winner.

Now imagine that the stock you selected was A. This would mean that Colin would tell you that B is the dog and give you the option of swapping. If you were to swap you would win (swap to C the winner) and if you were to stay put (on A) you would lose.

If you had chosen stock B then Colin would have told you stock A was the dog and given you the option to swap. If you were to swap (to C) then you would win and if you were to stay put (on B) you would lose.

Lastly if you had picked C then he would have either told you that A or B were the dog and given you the option to swap. Regardless of which one he called the dog you would lose if you swapped (to A or B) and win if you stayed put (on C).

So what this means is that if you had a policy of always swapping then you would win if you initially chose A or B and lose if you chose C. If you had a policy of staying put then you would win if you initially chose C but lose if you chose A or B. Given that you have an equal chance of picking all the stocks (1/3 chance to pick each one) you will have a 2/3 chance of picking A or B and a 1/3 chance of picking C. This means that if you swapped when given the option you would win 2/3 of the time (if you initially chose A or B) and lose 1/3 of the time (if you chose C). Therefore the answer is that you should always swap when given the option as it is statistically more likely that you have chosen A or B as your initial stock and it would result in swapping to C (the winner).