e k 1 n 2 15 in the game matching coins two people each flip a biased

Question

(E) (k+1)/(n+2)
15. In the game "matching coins" two people each flip a biased coin with probability of heads
p=1/3. If the two coins show the same face, person 1 wins and gets both coins, if they don't
person 2 wins and gets both coins. The probability that person 1 wins is closest to
(A) 1/4
(B) 5/9
(C) 2/3
(D) 1/2
(E) 4/9
16. The probability that a certain test gives a positive reaction to a rare disease is 99/100 but
the test is not perfect. The probability that the test gives a positive reaction eventhough the
patient does not have the disease is 1 in a million. Compute the probability that a patient with
a positive test actually has the disease if the prevalence of the disease is also 1 in a million.
(Le. the known proportion of people with the disease in the population is 10). Choose the
closest answer.
(A) 35.2%
(B) 99.0%
(C) 68.3%
(D) 49.8%
(E) 87.9%
17. Consider a box with (n-1) tickets with the number 0 and only one ticket with the number
1. Let p, be the probability of drawing (with replacement) (n-1) tickets with the number 0
followed by the ticket with the number 1. Then, when n is large, the value of np, is closest to
(A) 0.3678
(B) x
(C) 0.5
(D) 0.6105
(E) 0
18. In a box we put one million white balls but only two thousand black balls. We shake the box
and draw (without looking) four hundred balls from the box. Use the Poisson approximation
to the Binomial distribution to find the probability of drawing three black balls. The closest
answer is,
(A) 0.025
(B) 0.017
(C) 00035
(D) 0.001
(E) 0.038
19. Two coins with probability p of heads are tossed independently. If you learn that one of them
is heads; What's the probability that the other is also heads?
(A) P
(B)
(D)
(E)
20. Define new logical operations of multiplication and addition by: ab=a+b and a @ b = üb.
With these new definitions, is the following equation true?
(ab) (cd)(ab) (cd)
Simplify both sides and choose the most correct,