Suppose that an insurance company classifies people into one of three classes — good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1-year span are, respectively, .05, .15, and .30. If 20 percent of the population are “good risks,” 50 percent are “average risks,” and 30 percent are “bad risks,” what proportion of people have accidents in a fixed year? If policy holder A had no accidents in 1987, what is the probability that he or she is a good (average) risk?

If A, B, and C are mutually exclusive events with P(A) = 0.2, P(B) = 0.3, P(C) = 0.4, determine the following probabilities. a. P(A ∪ B ∪ C) b. P(A ∩ B ∩ C) c. P(A ∩ B) d. P[A ∪ B) ∩ C] e. P(A' ∩ B' ∩ C')

Let $A$ and $B$ be events with $P(A)=0.6, P(B)=0.3$, and $P(A \cap B)=0.2$. Find:

(a) $P(A \cup B)$; $\quad$ (b) $P(A \mid B)$; $\quad$ (c) $P(B \mid A)$.

Two percent of woman of age 45 who participate in routine screening have breast cancer. Ninety percent of those with breast cancer have positive mammographies. Ten percent of the women who do not have breast cancer will also have positive mammographies. Given a woman has a positive mammography, what is the probability she has breast cancer?

Let A, B, C be events such that P(A) = .2, P(B) = .3, P(C) = .4. Find the probability that at least one of the events A and B occurs if (a) A and B are mutually exclusive; (b) A and B are independent. Find the probability that all of the events A, B, C occur if (c) A, B, C are independent; (d) A, B, C are mutually exclusive.

Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a table. If the side facing up is red, what is the probability that the other side is also red?

What is the difference between the waterfall approach and agile project management?

Suppose that an insurance company insures its clients for flood damage to property. Can the company reasonably expect that the claims from its clients will be independent of each other?

The sample mean and sample variance of five data values are, respectively, $\bar{x}=104$ and $s^{2}=16$. If three of the data values are 102, 100, 105, what are the other two data values?

A system is composed of four components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector $(x_1, x_2, x_3, x_4)$ where $x_i$ is equal to 1 if component i is working and is equal to 0 if component i is failed. (a) How many outcomes are in the sample space of this experiment? (b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working. Specify all the outcomes in the event that the system works. (c) Let E be the event that components 1 and 3 are both failed. How many outcomes are contained in event E?

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

From a group of n people, suppose that we want to choose a committee of k,

$$k \leq n$$

, one of whom is to be designated as chairperson. (a) By focusing first on the choice of the committee and then on the choice of the chair, argue that there are

$$\left( \begin{array} { l } { n } \\ { k } \end{array} \right)$$

k possible choices. (b) By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are

$$\left( \begin{array} { c } { n } \\ { k - 1 } \end{array} \right) ( n - k + 1 )$$

possible choices. (c) By focusing first on the choice of the chair and then on the choice of the other committee members, argue that there are n

$$\left( \begin{array} { l } { n - 1 } \\ { k - 1 } \end{array} \right)$$

possible choices. (d) Conclude from parts (a), (b), and (c) that

$$k \left( \begin{array} { l } { n } \\ { k } \end{array} \right) = ( n - k + 1 ) \left( \begin{array} { c } { n } \\ { k - 1 } \end{array} \right) = n \left( \begin{array} { l } { n - 1 } \\ { k - 1 } \end{array} \right)$$

(e) Use the factorial definition of

$$\left( \begin{array} { l } { m } \\ { r } \end{array} \right)$$

to verify the identity in part (d).

Each of 2 balls is painted either black or gold and then placed in an urn. Suppose that each ball is colored black with probability

$$\frac { 1 } { 2 }$$

and that these events are independent. (a) Suppose that you obtain information that the gold paint has been used (and thus at least one of the balls is painted gold). Compute the conditional probability that both balls are painted gold. (b) Suppose now that the urn tips over and 1 ball falls out. It is painted gold. What is the probability that both balls are gold in this case? Explain.

On rainy days, Joe is late to work with probability .3; on nonrainy days, he is late with probability .1. With probability .7, it will rain tomorrow. (a) Find the probability that Joe is early tomorrow. (b) Given that Joe was early, what is the conditional probability that it rained?