The following exercise involves the Fundamental Counting Principle.

A restaurant offers the items listed in the table. How many different meals consisting of a main course, a drink, and a dessert can be selected at this restaurant?

Main courses	Drinks	Desserts
Chicken	Iced tea	Ice cream
Beef	Apple juice	Layer cake
Lasagna	Cola	Blueberry pie
Quiche	Ginger ale
	Coffee

To battle against spam, Bob installs two anti-spam programs. An email arrives, which is either legitimate (event $L$) or spam (event $L^c$), and which program $j$ marks as legitimate (event $M_j$) or marks as spam (event ${M_j}^c$ ) $j \in {1, 2}$. Assume that 10% of Bob’s email is legitimate and that the two programs are each “90% accurate” in the sense that $P(M_j |L) = P({M_j}^c |L^c) = 9/10$. Also assume that given whether an email is spam, the two programs’ outputs are conditionally independent. (a) Find the probability that the email is legitimate, given that the 1st program marks it as legitimate (simplify). (b) Find the probability that the email is legitimate, given that both programs mark it as legitimate (simplify). (c) Bob runs the 1st program and $M_1$ occurs. He updates his probabilities and then runs the 2nd program. Let $\widetilde{P}(A) = P(A|M_1)$ be the updated probability function after running the 1st program. Explain briefly in words whether or not $\widetilde{P}(L|M_2) = P(L|M_1\cap M_2)$: is conditioning on $M_1 \cap M_2$ in one step equivalent to first conditioning on $M_1$, then updating probabilities, and then conditioning on $M_2$?

The manufacturing of semiconductor chips produces 2% defective chips. Assume that the chips are independent and that a lot contains 1000 chips. Approximate the following probabilities: a. More than 25 chips are defective. b. Between 20 and 30 chips are defective.

A lot of 100 semiconductor chips contains 20 that are defective. Two are selected randomly, without replacement, from the lot. a. What is the probability that the first one selected is defective? b. What is the probability that the second one selected is defective given that the first one was defective? c. What is the probability that both are defective? d. How does the answer to part (b) change if chips selected were replaced prior to the next selection?

A committee of seven is to be chosen from a group of ten men and eight women. In how many ways can the committee be chosen using each of the following selection requirements?

(a) No restriction is placed on the number of men and women on the committee.

(b) The committee must have exactly four men and three women.

(c) Susie refuses to serve on the committee.

(d) At least five women must serve on the committee.

(e) At most two men can serve on the committee.

(f) The committee is to have a chairman, a vice chairman, a secretary, and four other members.

The following exercise involves finding the probability of the union of events.

A die is rolled, and the number showing is observed. Determine whether the events $E$ and $F$ are mutually exclusive. Then find the probability of the event $E \cup F$.

(a) $E$: The number is even.

$F$: The number is odd.

(b) $E$ : The number is even.

$F$: The number is greater than four.

Systolic blood pressures are approximately Normal with a mean of 120 and a standard deviation of 8.

a. What percentage of people have a systolic blood pressure above 130?

b. What is the range of systolic blood pressures for the middle 60 percent of the population?

c. What percentage of people have a systolic blood pressure between 120 and 130?

d. Suppose people with systolic blood pressures in the top 15 percent of the population have their blood pressures monitored more closely by health care professionals. What blood pressure would qualify a person for this additional monitoring?

In the Mona Lisa study of cardiovascular risk factors in the French population, systolic blood pressure of 4609 French adults averaged $134.4 \mathrm{mmHg}$ with a standard deviation of $20.6 \mathrm{mmHg}$.

Sketch and clearly label the sampling model of the mean systolic blood pressure levels of samples of size 4609 based on the 68-95-99.7 Rule.

Assume that the sample is taken fro ma large population and the correction factor can be ignored.
Systolic Blood Pressure Assume that the mean sustolic blood pressure of normal adults is $120$ milimeters of mercury $(\mathrm{mm} \mathrm{Hg})$ and the standard deviation is $5.6$. Assume the variable is normally distributed.
If an individual is selected, find the probability that the individual's pressure will be between $120$ and $121.8 \mathrm{~mm} \mathrm{Hg}$.

Assume that the sample is taken fro ma large population and the correction factor can be ignored.
Systolic Blood Pressure Assume that the mean sustolic blood pressure of normal adults is $120$ milimeters of mercury $(\mathrm{mm} \mathrm{Hg})$ and the standard deviation is $5.6$. Assume the variable is normally distributed.
If a sample of $30$ adults is randomly selected, find the probability that the sample mean will be between $120$ and $121.8 \mathrm{~mm} \mathrm{Hg}$.

Explain how to use a table showing z-scores and percentiles to determine the percentage of data items between two z-scores.

A group of data items and their mean are given. a. Find the deviation from the mean for each of the data items. b. Find the sum of the deviations in part ( a).

$$29,38,48,49,53,77; Mean = 49$$

A college professor had students keep a diary of their social interactions for a week. Excluding family and work situations, the number of social interactions of ten minutes or longer over the week is shown in the following grouped frequency distribution. Use this information to solve the exercises.

$$\begin{array}{|c|c|} \hline \text{Number of Social Interactions} & \text{Frequency}\\ \hline 0 -4 & 12\\ \hline 5 - 9 & 16\\ \hline 10 - 14 & 16\\ \hline 15 - 19 & 16\\ \hline 20 - 24 & 10\\ \hline 25 - 29 & 11\\ \hline 30 - 34 & 4\\ \hline 35 - 39 & 3\\ \hline 40 - 44 & 3\\ \hline 45- 49 & 3\\ \hline \end{array}$$

What is the class width?

A college professor had students keep a diary of their social interactions for a week. Excluding family and work situations, the number of social interactions of ten minutes or longer over the week is shown in the following grouped frequency distribution. Use this information to solve the exercises.

$$\begin{array}{|c|c|} \hline \text{Number of Social Interactions} & \text{Frequency}\\ \hline 0 -4 & 12\\ \hline 5 - 9 & 16\\ \hline 10 - 14 & 16\\ \hline 15 - 19 & 16\\ \hline 20 - 24 & 10\\ \hline 25 - 29 & 11\\ \hline 30 - 34 & 4\\ \hline 35 - 39 & 3\\ \hline 40 - 44 & 3\\ \hline 45- 49 & 3\\ \hline \end{array}$$

Among the classes with the smallest frequency, which class has the least number of social interactions?