Insurance risk groups
An insurance company divides customers into three risk categories: High (H), Medium (M), and Low (L). Assume the following distribution of customers: 70% are High risks, 20% are Medium risks, and
are Low risks. The probabilities of a customer filing an accident claim (C) are: for High, for Medium, and for Low. A customer is chosen at random. (a) What is the probability that the customer files a claim?
(b) If a customer has filed a claim, what is the probability that the customer was in the High risk category?
Passing the exam
An exam has 3 sections. The first section has 2 problems, the second section has 3 problems, and the third section has 4 problems.
To pass a section, you must solve at least 1 problem in that section. You need to pass all 3 sections to pass the exam.
The probability of solving any one problem is 0.75. The problems are not related.
What is the probability of passing the exam?
QC Inspector
A lot containing 10 components is sampled by a quality control inspector. The lot contains 7 good components and 3 defective components. A random sample of 2 is taken (without replacement) by the inspector. Let X represent the number of good components in the inspector’s sample.
(a) What is the probability mass function (PMF) of
? (b) What is the cumulative distribution function (CDF) of
? (c) Find
. (d) Find
.
Doctor’s visits
I am supposed to visit my doctor once every four months to get my blood pressure checked. However, I have a hard time sticking to the schedule, and as a result, rarely do I visit my doctor three times a year. The function below is the CDF of X, the number of my visits in a year.
(a) Find the probability that I will visit my doctor more than once in a year.
(b) Find the number of times I expect to visit my doctor in a typical year.
Potholes
There are a lot of small potholes on the US-1N between Conowingo, MD and Concordville, PA, and once or twice a month, Deep drives this stretch on my way to Philadelphia from Charlottesville. He has realized that this situation (potholes on the highway) can be modeled by a Poisson process, and that if the RV
denotes the distance between successive potholes (measured in miles), then the CDF of is: (a) What is the mean number of potholes in a 2-mile stretch of the highway?
(b) What is the probability that there will be at least 2 potholes in a 2-mile stretch of the highway?