HPSGeneral32010 - Probabilistic Induction Brain Teasers for...

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12/6/10 11 Probabilistic Induction Brain Teasers for Thursday v # 1.I have three coins in a hat, one is red on both sides, one blue on both sides, and one red on one side and blue on the other. A games master grabs one of the coins at random and plops it on the table. We see What is the probability the flip side is red? v #2. I have three children. I will tell you that at least one is a girl. Given this knowledge, what is the probability they are all girls? (assume 50:50 birth probs.) v #3. In one of three identical boxes is a big ruby, the others have live rabid rats. You pick box A. I promised to tease you by opening a box with as rat, so I open box B and out jumps a rabid rat. Would you like to change to box C? re d
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12/6/10 22 A Paradox involving Induction v Where do we seek evidence for an inductive belief? E.g. Consider the proposition: All rubies are red. v Modus Ponens (P Q & P)
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Unformatted text preview: we find rubies and check them for redness. v But the proposition is logically equivalent to All non-red things are non-rubies Note: P Q if and only if not-Q not-P This suggests that we find non-red things and Of form P Q 12/6/10 33 Backward Induction Paradox v Its Saturday and Judge sentences a killer as follows: ‘ You will be executed at noon in one of the next 7 days. I just wrote the day on this piece of paper. On the morning of the day you will be shown the paper and you will be surprised because you will not know ahead of time.’ v The killer is depressed but his lawyer says you have nothing to worry about- the judge can not carry out the terms of his sentence, “… clearly it can’t be next Sunday because you would know Saturday ahead of time (no surprise), therefore it can’t be Saturday because you would know Friday, etc....
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