[Suman Ganguli]

A reminder that we will have our first exam on Monday.

As I discussed in class on Wednesday, the best way to review is to look at exercises #1-11 on the final exam review sheet I handed out on Monday (except #5 and #7, since those deal with combinations and we didn’t have time to fully cover that concept)

The only additional material not covered in those exercises that may be on the exam is the material in Sec 2.5 that we covered–quartiles, interquartile range (IQR), percentiles, and boxplots. As I recommended, you should compute the quartiles and IQR for the dataset in exercise #1, and also draw the boxplot.

It’s also a good idea to review the vocabulary and examples on the daily handouts that we used in class.

[Suman Ganguli]

If you were absent on Wednesday, you can get a pdf of the final exam review sheet under the Files section:
http://openlab.citytech.cuny.edu/groups/math1272-statistics-fall2013-ganguli/files/

Unfortunately the pdf is a slightly older version of the review sheet and is missing some of the relevant exercises on the hard copy I handed out Wednesday.

Exercises 1-6 are the same, so do those. I will write out the additional exercises below, along with some hints on how to do them

[Suman Ganguli]

“A club elects a president, a vice-president, and a secretary. How many sets of officers are possible if there are 20 members and any member can be elected to each position? No person can hold more than one position.”

This is an exercise involving permutations, which we discussed in class on Wednesday, but is similar to counting exercises we discussed previously in the course.

In this case, if there are 20 possible choices for president, then there are 19 remaining choices for vice-president, and then 18 remaining choices for secretary. So there are 20*19*18 different sets of officers.

[Suman Ganguli]

This exercise also involves permutations:

“How many ways can gold, silver, and bronze medals be awarded for a race run by 8 people?”

By similar reasoning as above, there are 8*7*6 possible permuations for 1st-2nd-3rd.

[Suman Ganguli]

“How many five-digit zip codes can be made where all digits are unique? The possible digits are the numbers 0 through 9.”

Here, “all digits are unique” means no digit in the 5-digit code can be repeated. So there are 10 choices for the 1st digit, 9 remaining choices for the 2nd, 8 remaining choices for the 3rd, and so on.

So the answer is 10*9*8*7*6

For additional counting exercises similar to these, look at:

Sec 3.1: Examples 4 & 11
Sec 3.4: Examples 2, 3 & 6

[Suman Ganguli]

This exercise involves conditional probabilities in the context of an empirical dataset:

“Of a company’s employees, 35% are women and 8% are married women. Suppose an employee is selected at random. Given that the selected employee is a woman, what is the probability that she is married?”

For this question, consider the events
M = “the employee is married”
W = “the employee is a woman”

We want to find the conditional probability P(M | W). It should make intuitive sense that P(M | W) = 8/35.

Here’s how to make that precise:

Suppose the company has N total employees. To calculate P(M | W), the sample space consists of the 35% of employees who are women, i.e., the 0.35*N women employees. Within that sample space, 0.08*N are married (since 8% of employees are married women). So

P(M | W) = 0.08*N / 0.35*N = 0.08/0.35 = 8/35.

To review this concept of conditional probability in the context of empirical data, you should review Sec 3.2 Example 1, as well as the example on the Day 10 handout. For a similar exercise, try #19 in Sec 3.2.

[Suman Ganguli]

The last exercise to review on the final exam review sheet, #11:

“A student has applied to graduate schools of two universities, A and B. The student feels she has a 60% chance of receiving an offer from A, and a 50% chance of receiving an offer from B. If she receives an offer from B, she believes she has a 80% chance of receiving an offer from A.”

First, note that these probabilities are “subjective probabilities”, based on the student’s beliefs. Start by recording the given info as follows:

P(A) = 0.6
P(B) = 0.5
P(A | B) = 0.8 (translations of the last sentence, which implies that receiving offers from A and B are not independent–at least according to the student’s subjective beliefs)

Now here are the questions you are asked to answer:

(a) What is the probability that both universities will make her an offer? (i.e., what is P(A & B)?)
(b) What is the probability that at least one university will make her an offer? (i.e., what is P(A or B)?)
(c) If she receives an offer from B, what the probability that she will not receive an offer from A?” (i.e., what is P( not A | B)?

We can use the rules we learned in Ch3 for calculating these probabilities, in terms of the given values of P(A), P(B), and P(not A | B)

(a) Use the multiplication rule (Sec 3.2): P(A & B) = P(B) * P(A | B) = 0.5*0.8 = 0.4
(b) Use the addition rule (Sec : P(A or B) = P(A) + P(B) – P(A and B) = 0.6 + 0.5 – 0.4 = 0.7
(c) For this we use the “complement rule” (Sec 3.1), for A conditional on B: P( A | B) + P( not A | B) = 1, i.e., P( not A | B) = 1 – P( A | B) = 1 – 0.8 = 0.2

For this exam, you don’t have to memorize these rules–I will provide them to you. But you should understand this and similar examples of how they are applied.

In addition to this exercise, look at the following examples and exercises in the textbook:

Sec 3.2: Examples 3 & 5, “Try It Yourself” exercises 3 & 5
Sec 3.3: Examples 2 & 4, “Try It Yourself” exercises 2 & 4