In the 19th century, cavalries were still an import- ant part of the European military complex. While horses have many wonderful qualities, they can be dangerous beasts, especially if poorly treated. The Prussian army kept track of the number of fatal- ities caused by horse kicks to members of 10 of their cavalry regiments over a 20-year time span. If these fatalities occurred independently and with equal probability for each regiment, then the number of deaths by horse kick per regiment per year should follow a Poisson distribution. On the other hand, if some regiments during some years consisted of particularly bad horsemen,11 then the events would not occur with equal probability, in which case we would expect a frequency dis- tribution different from the Poisson distribution. The following table shows the data, expressed as the number of fatalities per regiment-year (Bort- kiewicz 1898).

a. What is the mean number of deaths from horse kicks per regiment-year?
b.   Test whether a Poisson distribution fits these data. (please break down each step of this as much as possible-- thank you so much!)

Probability theory is the formalism to study:

1. Choice
2. Chance
3. Risk
4. Criticality
5. All of the above

Match the following terms to a definition.

Frequency interpretation of probability

Equally likely interpretation of probability

Bayesian interpretation of probability

1. Degree of confidence or certainty based on knowledge of the phenomenon.
2. If a sample space Ω consists of a finite number of outcomes n, which are all equally likely to occur, then the probability of each simple event is 1/n.
3. The probability of an event is the limiting proportion of time the event occurs in a set of n repetitions of the experiment.

Decision theory is the formalism to study:

1. Criticality
2. Chance
3. Risk
4. Choice
5. Resilience

Together, probability and decision theory provide the formalism to study:

1. Uncertainty
2. Risk
3. Cost
4. Choice
5. None of the above

Match the following terms to a definition. [Hint: Modified text book Figure 1.2]

Risk identification

Risk impact assessment

Risk prioritization analysis

Risk mitigation

Ideal scenario identification

A. reassess existing risk events and identify new risk events

B. probabilities & consequences of risk events are assessed

C. knowing what needs to go right

D. risk events and their relationships are defined

E. assess risk criticality

Negative Binomial experiment is based on sequences of Bernoulli trials with probability of success p. Let x+m be the number of trials to achieve m successes, and then x has a negative binomial distribution. In summary, negative binomial distribution has the following properties

• Each trial can result in just two possible outcomes. One is called a success and the other is called a failure.
• The trials are independent
• The probability of success, denoted by p, is the same on every trial.
• The experiment consists of m successes, and x+m repeated trials, and the m^th success occurs at the (x+m)^th trial.

1. Write down the probability distribution P(x=k), consistent with the notation here
2. If you are tossing a regular coin repeatedly, what is the probability that the 3^rd head occurs at the 6^th time you toss it?
3. Anne is selling girl scot cookies in her neighborhood with 20 houses. She has a target to sell 10 boxes. Suppose each house has a probability 0.6 to buy one box of her cookies. What is the probability that she sells the last box at the 15^th house? What is the probability that she exhausts all 20 houses?

It has been said that Maslow's theory does not hold up globally; that is, other cultures do not see self-actualization as the highest form of need. Rather, some cultures believe service to others or family is the highest level of need fulfillment. What is your opinion on this? Explain.

The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Complete parts​ (a) through​ (d) below. LOADING... Click the icon to view the table. a. Find the probability of getting exactly 6 girls in 8 births. . 107 ​(Type an integer or a decimal. Do not​ round.) b. Find the probability of getting 6 or more girls in 8 births.

A process that fills packages is not stopped until four packages whose weight falls outside the specification are detected. Assume that each package has probability 0.01 of falling outside the specification and that the weights of the packages are independent.

a.Define the random variable of interest, its support, and parameter values over this period.

b. What is the probability that 50 packages need to be filled until the process is stopped?

c.Find the mean and variance of the number of packages that will be filled before the process is stopped.

In a multiuser wireless network the data frames are being transmitted when the probability of collision during the transmission is 0.75. The transmitters try to send the collision-free frames using up to 4 independent trials (transmission) as needed. The transmitters indefinitely discard the data frames after 4 failures. Calculate the probability that a data frame can be successfully transmitted (not be discarded). Let ? denote the random number of trails until the frame is transmitted without collision, find ?(? ≤ 5) =?

Define the random variable X to be the number of times in the month of June (which has 30 days) Susan wakes up before 6am

a. X fits binomial distribution, X-B(n,p). What are the values of n and p?

c. what is the probability that Susan wakes us up before 6 am 5 or fewer days in June?

d. what is the probability that Susan wakes up before 6am more than 12 times?

According to an​ article, 34​% of adults have experienced a breakup at least once during the last 10 years. Of 9 randomly selected​ adults, find the probability that the​ number, X, who have experienced a breakup at least once during the last 10 years is

a. exactly​ five; at most​ five; at least five.

b. at least​ one; at most one.

c. between 5 and 7 inclusive.

d. Determine the probability distribution of the random variable X.

1 An urn contains three green chips and four blue chips. Two chips are removed is succession. What is the probability both are blue if:

a.) The chips are sampled with replacement?  
b.) The chips are sampled without replacement?

2 An urn contains 5 orange and 4 red balls. Three balls are removed. Lex R be the random variable (R.V.) denoting the number of red balls in the sample of three balls. What is the probability two of the balls in the sample are red?