Introduction to Probability 1st Edition solutions

Introduction to Probability 1
Bobo, the amoeba from Chapter 2, currently lives alone in a pond. After one minute Bobo will either die, split into two amoebas, or stay the same, with equal probability. Find the expectation and variance for the number of amoebas in the pond after one minute.

Introduction to Probability 1
A book has n typos. Two proofreaders, Prue and Frida, independently read the book. Prue catches each typo with probability p1 and misses it with probability q1 = 1 p1, independently, and likewise for Frida, who has probabilities p2 of catching and q2 = 1p2 of missing each typo. Let X1 be the number o

Introduction to Probability 1
Let U Unif(0, 8). (a) Find P(U 2 (0, 2) [ (3, 7)) without using calculus. (b) Find the conditional distribution of U given U 2 (3, 7).

Introduction to Probability 1
In humans (and many other organisms), genes come in pairs. Consider a gene of interest, which comes in two types (alleles): type a and type A. The genotype of a person for that gene is the types of the two genes in the pair: AA, Aa, or aa (aA is equivalent to Aa). According to the Hardy Weinberg law,

Introduction to Probability 1
A group of 21 women and 14 men are enrolled in a medical study. Each of them has a certain disease with probability p, independently. It is then found (through extremely reliable testing) that exactly 5 of the people have the disease. Given this information, what is the expected number of women who h

Introduction to Probability 1
The Rayleigh distribution from Example 5.1.7 has PDF f(x) = xex2/2 , x Let X have the Rayleigh distribution. (a) Find P(1 X3). (b) Find the first quartile, median, and third quartile of X; these are defined to be the values q1, q2, q3 (respectively) such that P(X qj ) = j/4 for j = 1, 2, 3.

Introduction to Probability 1
Emails arrive in an inbox according to a Poisson process with rate 20 emails per hour. Let T be the time at which the 3rd email arrives, measured in hours after a certain fixed starting time. Find P(T > 0.1) without using calculus. Hint: Apply the count-time duality.

Introduction to Probability 1
People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives there are no two people with the same birthday, but when person

Introduction to Probability 1
How many ways are there to permute the letters in the word MISSISSIPPI?

Introduction to Probability 1
Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day. (a) What is the probability that Carl arrives first? For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition o

Introduction to Probability 1
One of two doctors, Dr. Hibbert and Dr. Nick, is called upon to perform a series of n surgeries. Let H be the indicator r.v. for Dr. Hibbert performing the surgeries, and suppose that E(H) = p. Given that Dr. Hibbert is performing the surgeries, each surgery is successful with probability a, independ

Introduction to Probability 1
Fred wants to travel from Blotchville to Blissville, and is deciding between 3 options (involving dierent routes or dierent forms of transportation). The jth option would take an average of j hours, with a standard deviation of j hours. Fred randomly chooses between the 3 options, with equal probabil

Introduction to Probability 1
A circle with a random radius R Unif(0, 1) is generated. Let A be its area. (a) Find the mean and variance of A, without first finding the CDF or PDF of A. (b) Find the CDF and PDF of A.

Introduction to Probability 1
There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (in a chess game, one player has the white pieces and the other player has

Introduction to Probability 1
A spam filter is designed by looking at commonly occurring phrases in spam. Suppose that 80% of email is spam. In 10% of the spam emails, the phrase free money is used, whereas this phrase is only used in 1% of non-spam emails. A new email has just arrived, which does mention free money. What is the

Introduction to Probability 1
(a) Make up a PDF f, with an application for which that PDF would be plausible, where f(x) > 1 for all x in a certain interval. (b) Show that if a PDF f has f(x) > 1 for all x in a certain interval, then that interval must have length less than 1.

Introduction to Probability 1
A certain course has a freshmen, b sophomores, c juniors, and d seniors. Let X be the number of freshmen and sophomores (total), Y be the number of juniors, and Z be the number of seniors in a random sample of size n, where for Part (a) the sampling is with replacement and for Part (b) the sampling i

Introduction to Probability 1
People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives there are no two people with the same birthday, but when person

Introduction to Probability 1
A fair coin is flipped twice. Let X be the number of Heads in the two tosses, and Y be the indicator r.v for the tosses landing the same way. (a) Find the joint PMF of X and Y . (b) Find the marginal PMFs of X and Y . (c) Are X and Y independent? (d) Find the conditional PMFs of Y given X = x and of

Introduction to Probability 1
A family has 3 children, creatively named A, B, and C. (a) Discuss intuitively (but clearly) whether the event A is older than B is independent of the event A is older than C. (b) Find the probability that A is older than B, given that A is older than C.