Discrete Probability Distribution/Bayes Law. Consider the binomial distribution:

Question

Discrete Probability Distribution/Bayes Law. Consider the binomial distribution: the chance of getting M "heads" #4. out of N coin tosses is (NUIMN-(I-p) where p is the probability of a single "head" when tossing a coin once problem you did in class. the reverse Show that the posterior probability that Prof. Norman chose Um I is the same as Prof. O'Melia's in the following Imagine that there are two urns of balls. Um I has 25% red balls and 75% blue balls, while Urn II has Prof. O'Melia picks an urn at random and draws four balls from that urn (replacing each ball after drawing it). Three out of the four balls he draws are blue. Prof. Norman picks an urn at random and draws twelve balls (again with replacement). Seven of the twelve are blue a. b. Ve

Explanation / Answer

Pr( Urn 1 , Red) = 0.25

Pr(Urn 1, Blue) = 0.75

Pr( Urn 2, Red) = 0.75

Pr(Urn 2, BLue) = 0.25

(a) An Urn at random and draws four balls from that urn.

Pr( 3 out of 4 balls are blue) = Pr(Urn 1) * Pr(3 blue balls out of 4 balls) + Pr(Urn 2) * Pr(3 blue out of 4 balls)

1/2 * 4C3 * (0.75)3 (0.25) + 1/2 * 4C3 * (0.75)(0.25)3

= 0.2109 + 0.0235

= 0.2344

Pr(Urn 1 l 3 out of 4 balls are blue) = 0.2109/0.2344 = 0.9

(b) Pr(7 out of 12 balls) = Pr(Urn 1) * Pr(7 blue balls out of 12 balls) + Pr(Urn 2) * Pr(7 blue out of 12 balls)

= 1/2 * 12C7 * (0.75)3 (0.25) + 1/2 * 12C7 * (0.75)(0.25)3

= 0.0058 + 0.0517

= 0.0575

Pr(Urn 2 l 3 out of 4 balls are blue) = 0.0517/0. 0575= 0.9

so we get the same probability for both

so both professor will have same chances of getting Urn 1.

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