I said I could change my mind and I have. The misdirection in the wording of the problem (and several of the explanations of the answer) is the focus on the participant rather than the host. The host captures the benefit of the 2/3 odds of having the car in the first stage since he gets two of the three choices. Once he has seen the first stage outcome, he always discards a loser and then in the 2 of 3 first stage outcomes where he has the winner, he keeps it. So in 2 out of 3 first stage outcomes, the host has kept the winner. The participant, not knowing the first stage outcome, can elect to swap his 1 in 3 chance of having won the first stage for the host's 2 in 3 chance of having the first stage winner.Kwinzenried (talk) 16:10, 7 January 2012 (UTC)