Oops, I only cut-and-pasted half of what I wanted. This comes after my other reply.
Yes, these sorts of problems can be confusing. But the confusion is propagated by educators who refuse to recognize that what they asked is not what they intended to ask, and so they provide inconsistent answers.
Say you are on a game show, and pick Door #1. The host opens door #3 to show that it does not have the prize, and offers to let you switch to door #2. Should you? Most people will initially reason that door #3 is prize-less 2/3 of the time, evenly split between cases where the prize is behind door #1 and door #2. So it would be pointless to switch. But that is wrong. Few educators will explain why by solving the problem rigorously. They will use an analogy like pointing out how the original choice is right only 1/3 of the time, and since the host can always open a prize-less door, that can’t change.
People don’t believe these educators because their 1/2 answer is indeed more rigorous than the analogy. It just makes a mistake. The probabilities to use are not the probabilities that the cases exist, but the probabilities that the observed result would occur. The existence probabilities are the same, but the probability of the observed result when the initial door was correct is half of what it is when the initial choice was incorrect.
Yes, these sorts of problems can be confusing. But the confusion is propagated by educators who refuse to recognize that what they asked is not what they intended to ask, and so they provide inconsistent answers.
Say you are on a game show, and pick Door #1. The host opens door #3 to show that it does not have the prize, and offers to let you switch to door #2. Should you? Most people will initially reason that door #3 is prize-less 2/3 of the time, evenly split between cases where the prize is behind door #1 and door #2. So it would be pointless to switch. But that is wrong. Few educators will explain why by solving the problem rigorously. They will use an analogy like pointing out how the original choice is right only 1/3 of the time, and since the host can always open a prize-less door, that can’t change.
People don’t believe these educators because their 1/2 answer is indeed more rigorous than the analogy. It just makes a mistake. The probabilities to use are not the probabilities that the cases exist, but the probabilities that the observed result would occur. The existence probabilities are the same, but the probability of the observed result when the initial door was correct is half of what it is when the initial choice was incorrect.
Yes, these sorts of problems can be confusing. But the confusion is propagated by educators who refuse to recognize that what they asked is not what they intended to ask, and so they provide inconsistent answers.
Say you are on a game show, and pick Door #1. The host opens door #3 to show that it does not have the prize, and offers to let you switch to door #2. Should you? Most people will initially reason that door #3 is prize-less 2/3 of the time, evenly split between cases where the prize is behind door #1 and door #2. So it would be pointless to switch. But that is wrong. Few educators will explain why by solving the problem rigorously. They will use an analogy like pointing out how the original choice is right only 1/3 of the time, and since the host can always open a prize-less door, that can’t change.
People don’t believe these educators because their 1/2 answer is indeed more rigorous than the analogy. It just makes a mistake. The probabilities to use are not the probabilities that the cases exist, but the probabilities that the observed result would occur. The existence probabilities are the same, but the probability of the observed result when the initial door was correct is half of what it is when the initial choice was incorrect.