Monty Hall

This is a continuation of the previous posts under this title.
Because probability is a branch of mathematics, there's a natural assumption that there must always be a definitive "correct" answer to any problem. But this isn't so. Probability is an assessment of the likelihood that something will, or will not occur based on the information available. And it's these last five words that are important. Different information may well - very probably will - produce a different assessment. Let’s look at an example. Suppose I’m giving a talk to a group of people on probability, and I’m going to use a pack of cards for demonstration. I shuffle the pack, and then ask what the probability is that the top card of the pack is a Jack. Unsurprisingly, the general consensus is that it’s 4/52 - 52 cards in the pack, and four of them are Jacks. But now, let’s suppose that, as I was shuffling the pack, I was rather careless, and one member of the group (call him or her X) caught a glimpse of the bottom card, and noted that it was a Jack. As far as X is concerned then, there are only 51 cards that could be on top, and only three of them are Jacks. They will therefore calculate the probability that the top card is a Jack as 3/51. Let us further suppose that another member of the group - Y - came into the room earlier on, and seeing the pack of cards on the table, decided to play a trick, and removed all the Aces. They will therefore know that the pack does not contain 52 cards, but only 48, and (not knowing about the bottom card) will calculate the probability that the top card is a Jack as 4/48. Let us further suppose that Z is a friend of Y, and knows what Y has done. Furthermore, Z is sitting next to X, who has just whispered the news about the bottom card. Z therefore knows that there are only 47 cards that the top card could be, and that only three of them are Jacks, and will therefore calculate the probability as 3/47.
I could go on, but I’m sure you get the point - X, Y and Z are all in possession of information not available to the rest of the group, and indeed they are all in possession of different information from each other, and have therefore come up with different probability figures. It is tempting to ask which of them is right, but in fact this is a meaningless exercise. They are all “right” in the sense that they have all made the correct mathematical calculation based on what they know. It is equally fatuous to ask what the “real” probability is - to the extent that such a thing exists, the real probability of the top card being a Jack is either 100% or 0% - it’s either a Jack or it isn’t.
Next time we'll look at how this affects you and Contestant II.