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Monty Hall, Vacant Spaces and Fire & Ice

The current pandemic period has produced some intriguing intellectual discussions here on BW. The Vacant spaces, Monty Hall and Fire and Ice articles are my favorite.

In some of the comments I’ve detected a certain confusion regarding (1) prior probabilities and (2) restricted choice. Both issues are, however, pretty simple. Let's set the record straight. 

(1). We all know that the probability of a 3-3 break is 36%. But it’s based on 13 unknown cards in each of opponents hands. Do we really know that the number is close to 50% once there are only nine cards in each hand?

(2). The principle of restricted choice seems to confuse. The easy way to deal with it is to realize that a play of a particular card decreases the probability its player holds any equivalent card. So, if an opponent has QJ, and plays one of them, it’s more unlikely that he does not have the other. Very simplistic, I agree, but roughly 65% true (Borel did the math in 1939).

The Monty Hall brain teaser is fascinating. I was an undergraduate math student when Selvin’s article was published. We laughed endlessly, as we thought the issue was so trivial. But it wasn’t. In particular, for us bridge players, the wisdom is simple: It is based on the deeply rooted intuition that revealing information that is already known does not affect probabilities. And you’ll never see Zia, Joe Grue or Disa ignore that intuition.

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