Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. If the coin lands on heads 525 or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than 525 times, then we shall conclude that it is a fair coin. If the coin is actually fair, what is the probability that we shall reach a false conclusion? What would it be if the coin were biased?
Suppose that in Problem 9.2, Al is agile enough to escape from a single car, but if he encounters two or more cars while attempting to cross the road, then he is injured. What is the probability that he will be unhurt if it takes him seconds to cross? Do this exercise for s = 5, 10, 20, 30.
Cars cross a certain point in the highway in accordance with a Poisson process with rate λ = 3 per minute. If Al runs blindly across the highway, what is the probability that he will be uninjured if the amount of time that it takes him to cross the road is s seconds? (Assume that if he is on the highway when a car passes by, then he will be injured.) Do this exercise for s = 2, 5, 10, 20.
Each member of a population of size n is, independently, female with probability or male with probability 1 – p. Let X be the number of the other n – 1 members of the population that are the same sex as is person 1. (So X = n – 1 if all people are of the same sex.)
a. Find P(X = i), i = 0, …, n – 1.
Now suppose that two people of the same sex will, independently of other pairs, be friends with probability α; whereas two persons of opposite sexes will be friends with probability β. Find the probability mass function of the number of friends of person 1.
Repeat Problem 3.87 when each of the 3 players selects from his own urn. That is, suppose that there are 3 different urns of 12 balls with 4 white balls in each urn.
An urn contains 12 balls, of which 4 are white. Three players—A, B, and successively C draw from the urn, A first, then B, then C, then A, and so on. The winner is the first one to draw a white ball. Find the probability of winning for each player if
a. Each ball is replaced after it is drawn;
b. The balls that are withdrawn are not replaced.
An urn contains 12 balls, of which 4 are white. Three players A, B, and C successively draw from the urn, A first, then B, then C, then A, and so on. The winner is the first one to draw a white ball. Find the probability of winning for each player if
a. Each ball is replaced after it is drawn:
b. The balls that are withdrawn are not replaced;