An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and
an opponent take turns selecting at random a single ball from the urn
without replacement. The person who selects the third WIN ball wins the
game. It does not matter who selected the first two WIN balls.
(a) If you draw first, find the probability that you win the game on your
fourth draw. (5%)
(b) If you draw first, what is the probability that you win? (10%)
Each bag in a large box contains 25 tulip bulbs. Three of fourths of the
bags contain bulbs for 10 red and 15 yellow tulips; one fourth of the bags
contain bulbs for 5 red and 20 yellow tulips. A bag is selected at random
and one bulb is selected randomly and planted. Given the probability that
it will produce a
(a) Red tulip, say P(R) . (5%)
(b) Yellow tulip, say P(Y ) . (5%)
(c) If the tulip is yellow, find the conditional probability that a bag having
10 red and 15 yellow tulips was selected. (5%)
In the casino game High-Low, there are three possible bets. Assume that
$1 is the size of the bet. A pair of fair six-sided dice is rolled and their sum
is calculated. If you bet low, you win $1 if the sum of the dice is {2, 3, 4, 5,
6}. If you bet high, you win $1 if the sum of the dice is {8, 9, 10, 11, 12}. If
you bet on {7}, you win $4 if a sum of 7 is rolled. Find the expected value of
the game to the bettor for each of these bets. (15%)
In ROC Lottery game, a player selects 6 integers out of the first 49 positive
integers. The Lotto company then randomly selects 6 and one special out of
the first 49 integers. Cash prizes are given to a player who matches 3, 4, 5,
or 6 integers with the selected 6 integers, or a player who not only match 3,
4, or 5 integers with the selected 6 integers, but also matches the one
special integer. Let X equal the number of integers selected by a player
that match the 6 selected integers, and Y equal the number of integer
selected by a player that match the one special integer.
(a) State the p.d.f. f (x, y) . (6%)
(b) Calculate the probability for each cash prize. (14%)
A seed distributor claims that 70% of its beet seeds will germinate. How
many seeds must be tested for germination in order to estimate p, the true
proportion that will germinate, so that the maximum error of the estimate
is ε = 0.03 with 95% confidence? (5%)
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