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Probability, Counting, Monty Hall Problem, Probability refs, Birthday…
Probability
Events
Independent
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2 events A and B where A is a subset of sample space S1 and B is a subset of a different sample space S2. Clearly, if A occurs then this won’t have any effect on occurrence of B and vice versa. Therefore A and B are independent events.
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Monty Hall Problem
3 Options: A, B, C;
One option has got a prize;
You choose some option (e.g. A);
Monty (game master) shows you option B;
Should you change to option C?
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After you chose one option (e.g. A),
P(Monty showing B) = P(Monty showing C) = 1/2;
P(Monty showing B | prize = A) = P(Monty showing C | prize = A) = 1/2;
P(Monty showing B | prize = B) = 0, P(Monty showing C | prize = B) = 1;
P(Monty showing B | prize = C) = 1, P(Monty showing C | prize = C) = 0;
Implement Bayes Formula.
H1: prize = A; H2 : prize = B; H3 : prize = C
Data = Monty showing C
P(H1 | D) = P(D|H1) P(H1) / P(D) = (1/2 * 1/3) / 1/2 = 1/3;
P(H2 | D) = P(D|H2) P(H2) / P(D) = 1 * 1/3 / 1/2 = 2/3;
P(H3 | D) = 0;
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Birthday problem
How many people do you need in a group to have 50% chance that 2 people will share the same birthday?
