3. Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip an

Question

3. Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 ofeach. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of Bowl #1? 4. In horse racing, one can make a trifecta bet by specifying which horse will come in first, which will come in second, and which will come in third, in the correct order. One can make a box trifecta bet by specifying which three horses will come in first, second, and third, without specifying the order. a. In an eight-horse field, how many different ways can one make a trifecta bet? b. In an eight-horse field, how many different ways can one make a box trifecta bet?

Explanation / Answer

Solution:

3) First the hypotheses:
A: the cookie came from Bowl #1
B: the cookie came from Bowl #2

And the priors:
P(A) = P(B) = 1/2

The evidence:
E: the cookie is plain

And the likelihoods:
P(E|A) = prob of a plain cookie from Bowl #1 = 3/4 = 0.75
P(E|B) = prob of a plain cookie from Bowl #2 = 1/2

Plug in Bayes's theorem and get
P(A|E) = P(A).P(E|A)/P(A)(E|A) + P(B).P(E|B) = [1/2.3/4]/[1/2.3/4 + 1/2.1/2 ] = 3/5
Probability of picking Bowl#1 given that the cookie turnout to be plain = 3/5

4)

a) In the horse racing, one can make a trifecta bet by specifying which horse will come in first, which will come second, and which will come in third. in the correct order.
The number of ways to make a trifecta bet is the number of permutations of 3 horses chosen from 8 in a specific order is
8!/(8-3)! = 8!/5! = 6*7*8 = 336  

b) In the horse racing, one can make a box trifecta bet by specifying which horse will come in first, which will come second, and which will come in third, without specifying the order. The number of ways to make a box trifecta bet is the number of combinations of 3 horses chosen from 8 without any specific order

8C3 = 8!/3!(8-3)!
= 56

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