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Chapter 3: Q.3.3 (page 97)
Use Equation to compute in a hand of bridge the conditional probability that East has spades given that North and South have a combined total of spades.
The conditional probability is
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In Problem 3.66a, find the conditional probability that relays and are both closed given that a current flows from to .
(a) An urn containswhite and black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability , they are all white. Hint: Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn.
(b) A pond containsdistinct species of fish, which we will call the Red, Blue, and Greenfish. There are Red, Blue, and Greenfish. Suppose that the fish are removed from the pond in random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Redfish are the first species to become extinct in the pond?
Hint: Write , and compute the probabilities on the right by first conditioning on the last species to be removed.
An engineering system consisting of n components is said to be a -out-of-system if the system functions if and only if at least of the components function. Suppose that all components function independently of one another.
(a) If the ith component functions with probability, compute the probability that a -out-of-system functions.
(b) Repeat part (a) for a -out-of-
system
A red die, a blue die, and a yellow die (all six sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die, which is less than that appearing on the red die. That is, with B, Y, and R denoting, respectively, the number appearing on the blue, yellow, and red die, we are interested in P(B < Y < R).
(a) What is the probability that no two of the dice land on the same number?
(b) Given that no two of the dice land on the same number, what is the conditional probability that B < Y < R?
(c) What is P(B < Y < R)?
Die A has 4 red and 2 white faces, whereas die B has
2 red and 4 white faces. A fair coin is flipped once. If it
lands on heads, the game continues with die A; if it lands on tails, then die B is to be used.
(a) Show that the probability of red at any throw is 12
(b) If the first two throws result in red, what is the probability of red at the third throw?
(c) If red turns up at the first two throws, what is the probability
that it is die A that is being used?
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