I.
Three horses A, B, and C are in a race. A is twice as likely
to win as B, and B is three times as loikely to win as C. What's
their respective probability of winning?
II.
A gambler loads a die so that the probability of turning up a number
of points is directly proportional to the number of points (i.e.,
2Pr(1)=Pr(2), etc.). Find Pr(6).
III.
Flip a coin loaded so that Pr(H)=2Pr(T). If you get H, then randomly
choose a number between 1 and 8 (included). If you get T, then randomly
choose a number between 1 and 5 (included). What's the probability
that you choose an even number?
IV.
John is given two vaccines, A and B. They are for the same disease and
act independently. The probability that A is successfull is 20%; that
probability that B is successful is 40%. One successful vaccine is enough
for vaccination. What's the probability that John is successfully vaccinated?
V.
You are given two urns, A and B. A contains 4 red marbles, 2
white marbles, and 4 blue marbles. B contains 2 red marbles and 3
white marbles. Toss a fair die; if 2 or 5 appear, then choose
a marble from B, otherwise choose a marble from A. What's the probability
that you choose:
1. a red marble
2. a white marble
3. a blue marble
VI.
You have a box with three coins, one fair, one two headed, and one
such that Pr(T)=2/3. Randomly choose a coin from the box and flip
it. What's Pr(H)? What's Pr(T)?
VII.
You have a bag with 1 red marble and 3 black marbles; you also have
a fair coin and a coin loaded so that Pr(H)=3Pr(T). Choose a marble
randomly. If it's red, then flip the fair coin; if it's black, flip
the loaded coin. If you get H, then flip the other coin; if you get
T, flip the same coin. Determine:
1. Pr(H on second flip)
2. Pr(T on both flips)
3. Pr(H on both flips)