Remember 1 we assume we have picked door a POA12 if the car is behind a then

Remember 1 we assume we have picked door a poa12 if

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Remember 1: we assume we have picked door a. P(O|A)=1/2 (if the car is behind a, then Monty opens b or c.) P(O|B)= 0 (if the car is behind b, then Monty never opens b.) P(O|C)=1 (if the car is behind c, then Monty will open b.) What is left is the unconditional probability P(O).

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The Monty Hall problem (7) To calculate P(O), note that the prize is only behind one door (i.e., A, B, C are disjoint events), hence: P(O)= P(O and A) + P(O and B) + P(O and C) The general multiplication rule gives: P(O)=P(A)*P(O|A)+P(B)*P(O|B)+P(C)*P(O|C) P(O)=1/3 * ½ +1/3* 0 +1/3* 1 P(O)=1/2
The Monty Hall Problem (8) Insert the numbers into the definition of conditional probability: That’s it! So, when you have picked door #1, always switch to door #3 when door #2 is opened.

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Back to last week: MC-answer guesses Last week, we have guessed MC- answers. By calculating the sample space and calculating the number of possibilities for a certain outcome we were (in principle) able to calculate the probability of an event. E.g. S=4^8=65536 P(X=8) = 1/65536 The probabilities of these events can be calculated in a more efficient way using the binomial setting.
Binomial probabilities (1) Binomial distributions are models for some categorical variables, typically representing the number of successes in a series of n trials. The observations must meet these requirements: The total number of observations n is fixed in advance. The outcomes of all n observations are statistically independent. Each observation falls into just one of two categories: success and failure. All n observations have the same probability of success, p .

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Binomial probabilities (2) The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p: B ( n,p ). The parameter n is the total number of observations. The parameter p is the probability of success on each observation. The count of successes X can be any whole number between 0 and n .
MC-answer guesses (once more) Binomial probabilities can be calculated as follows: So, in the coin toss example of last week: X 0 1 2 3 4 5 6 7 8 P(X) 0.10 0.27 0.31 0.21 0.09 0.02 0.003 0.0003 0.0000 2 k n k p p n k k X P ) 1 ( ) ( 31 . 0 75 . 0 * 25 . 0 * 2 8 ) 2 ( 6 2 X P

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Finding binomial probabilities: tables You can also look up the probabilities for some values of n and p in Table C in the back of the book. The entries in the table are the probabilities P ( X = k ) of individual outcomes. The values of p that appear in Table C are all 0.5 or smaller. When the probability of a success is greater than 0.5, restate the problem in terms of the number of failures.
Normal approximation If n is large, and p is not too close to 0 or 1, the binomial distribution can be approximated by the normal distribution N ( µ = np, σ 2 = np (1 p )) Practically, the Normal approximation can be used when both np ≥10 and n (1 p ) ≥10.

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• Fall '17
• prosecutor, Monty Hall, Monty

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