Example 2.8
Suppose that an airplane engine will fail, when in flight, with probability 1−p independently from engine to engine; suppose that the airplane will make a successful flight if at least 50 percent of its engines remain operative. For what values of p is a four-engine plane preferable to a two-engine plane? Read more
Example 2.9
Suppose that a particular trait of a person (such as eye color or left handedness) is classified on the basis of one pair of genes and suppose that d represents a dominant gene and r a recessive gene. Thus a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd is hybrid. The pure dominance and the hybrid are alike in appearance. Children receive one gene from each parent. If, with respect to a particular trait, two hybrid parents have a total of four children, what is the probability that exactly three of the four children have the outward appearance of the dominant gene? Read More
Example 2.10
Suppose that the number of typographical errors on a single page of this book has a Poisson distribution with parameter \(\lambda = 1\). Calculate the probability that there is at least some error on this page. Read More
Example 2.12 (\(\alpha\)-particles)
Consider an experiment that consists of counting the number of \(\alpha\)-particles given off in a one-second interval by one gram of radioactive material. If we know from past experience that, on the average, 3.2 such \(\alpha\)-particles are given off, what is a good approximation to the probability that no more than two α-particles will appear? Read More
Example 2.18
(Expectation of a Geometric Random Variable) Calculate the expectation of a geometric random variable having parameter p. Read More
Example 2.31
At a party \(N\) men throw their hats into the center of a room. The hats are mixed up and each man randomly selects one. Find the expected number of men who select their own hats.Read More
Example 2.32
Suppose there are 25 different types of coupons and suppose that each time one obtains a coupon, it is equally likely to be any one of the 25 types. Compute the expected number of different types that are contained in a set of 10 coupons. Read More
Example 2.37
(Sums of Independent Poisson Random Variables) Let \(X\) and \(Y\) be independent Poisson random variables with respective means \(\lambda_{1}\) and \(\lambda_{2}\). Calculate the distribution of \(X + Y\). Read More
Example 2.49
Suppose we know that the number of items produced in a factory during a
week is a random variable with mean 500.
1.What can be said about the probability that
this week’s production will be at least 1000?
2.If the variance of a week’s production is
known to equal 83333.3, then what can be said about the probability that this week’s
production will be between 400 and 600?
Read More
Example 2.52
The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation \(20^{\dagger}\). hours A battery is used until it fails, at which point it is replaced by a new ne. Assuming a stockpile of 25 such batteries, the lifetimes of which are independent, approximate the probability that over 1100 hours of use can be obtained.Read More
Example 2.53
Consider a particle that moves along a set of \(m + 1\) nodes, labeled 0, \(1, ..., m\), that are arranged around a circle. At each step the particle is equally likely to move one position in either the clockwise or counterclockwise direction. That is, if \(X_{n}\) is the position of the particle after its nth step then \(P(X_{n+1}=i+1|X_{n}=i)\) = \(P(X_{n+1}=i-1|X_{n}=i)\) = \(\frac{1}{2}\) where \(i + 1 \equiv 0\). When \(i = m\), and \(i − 1 \equiv m\) when \(i = 0\). Suppose now that the particle starts at 0 and continues to move around according to the preceding rules until all the nodes \(1, 2, ..., m\) have been visited. What is the probability that node \(i = 1, 2, ..., m\) is the last one visited?Read More
Sheldon Ross 10: Example 2.002
Let \(X\) represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of \(X\)?Read More
Sheldon Ross 10: Exercise 2.028
Suppose that we want to generate a random variable \(X\) that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p. Consider the following procedure:
- Flip the coin, and let 01, either heads or tails, be the result
- Flip the coin again, and let 02 be the result
- If 01 and 02 are the same, return to step 1
- If 02 is heads, set \(X = 0\), otherwise set \(X = 1\)