# STAT 3100 Study Guide - Midterm Guide: Binomial Theorem, Standard Deviation, Pairwise Independence

22 pages68 viewsFall 2014

School

University of GuelphDepartment

StatisticsCourse Code

STAT 3100Professor

Jeremy BalkaStudy Guide

MidtermThis

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Midterm Review Questions

1 Counting Rules

1. A certain long drive champion carries 4 diﬀerent drivers, 8 types of ball, and 5

diﬀerent types of tee. How many diﬀerent equipment choices can they make for

any one drive?

2. A tourist is about to take a cab to dinner then go to a show. They have a choice

of 3 diﬀerent cab companies, 8 diﬀerent restaurants, and 3 diﬀerent shows. How

many diﬀerent ways can they go about their evening?

3. A multiple choice test contains 25 multiple choice questions and 10 True/False

questions. 15 of the multiple choice questions have 5 answer options, and 10 have

4 answer options. How many diﬀerent ways are there of the student answering the

35 questions on this test?

4. In a bingo game, there are 75 numbers balls (numbered 1–75), and balls are drawn

without replacement until participants win all of the prizes.

(a) How many diﬀerent ways are there of choosing the ﬁrst 4 balls:

i. If we consider the order they are drawn important?

ii. If we do not consider the order they are drawn important?

(b) What is the probability the ﬁrst 5 balls contain the balls numbered 2 and 3?

(c) How many ways are there of dividing the balls into 5 diﬀerent groups of equal

size?

5. A class of 60 students is scheduled to write a midterm exam. They will be assigned

to write in 3 classrooms, one with 15 students, one with 20 students, and one with

25 students. How many diﬀerent ways are there of assigning these students to the

rooms?

1

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6. A class of 60 students is scheduled to write a midterm exam in which there are

two versions (yellow and white). How many diﬀerent ways can the 60 students be

assigned to the diﬀerent versions?

7. In the game of bridge, the cards in a 52 card deck are distributed randomly to 4

players (13 to each player).

(a) How many diﬀerent ways are there to deal out the 4 hands (the order in which

each person receives their cards is not important.)

(b) How many diﬀerent ways are there to deal out the 4 hands (the order in which

each person receives their cards is not important.)

(c) What is the probability that all 4 players get dealt 13 card hands that are all

of the same suit?

(d) Joe is playing bridge with friends. What is the probability that his 13 card

hand contains 3 sets of four of a kind?

8. Twenty friends are about to take a car trip in 4 rental cars. Paul, Sandy, Laela,

and Lilly will drive. Joe and Sally refuse to drive with Sandy, and George must

go in Lilly’s car. How many diﬀerent ways can the 20 friends be assigned to the

diﬀerent cars?

9. Eight friends are about to have dinner at a round table. Aahil and Pete refuse to

sit next to Joan. How many diﬀerent ways can the 8 friends be arranged around

the table, if only their relative position in of importance?

(a) How many diﬀerent ways can the 8 friends be arranged around the table, if

only their relative position in of importance?

(b) Aahil and Pete refuse to sit next to Joan. How many diﬀerent ways can the

friends be arranged around the table?

10. How many diﬀerent “words” (orderings of the letters) are possible for the letters in

the word version:

(a) If all the letters must be used (each letter must be used once and only once).

(b) If only 3 of the letters will be used (only 3 letter words are created).

(c) If the words can be of any length (the words can be from 1–7 letters long).

11. How many diﬀerent “words” (orderings of the letters) are possible for the letters in

the word classrooms:

(a) If all the letters must be used.

(b) If only 3 of the letters will be used.

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(c) If all the letters must be used and the words must start with cl and end with

om.

12. (a) What is the coeﬃcient of x3y5in the expansion of (x+y)8?

(b) What is the coeﬃcient of x3y5z8in the expansion of (x+y+z)16?

(c) What is the coeﬃcient of x3

1x8

2x2

3x9

4in the expansion of (x1+x2+x3+x4)22?

13. Show that:

(a) n

r=n

n−r

(b) nn−1

k−1=kn

k

(c) n

r=n

n−rn−1

r

14. Prove the following identities. (Hint: Use the binomial theorem.)

(a)

n

X

r=0 n

r= 2n

(b)

n

X

r=0

(−1)rn

r= 0

15. True or false:

(a) If k > r then n

k>n

r.

(b) If k > r then n

k<n

r.

(c) nPr≤nCr

(d) n

n=n

0

(e) nPn=nP0

2 Probability

16. What is the relative frequency interpretation of probability?

17. Suppose we are about to draw 3 cards without replacement from a standard 52

card deck.

(a) Give a listing of the elements of an appropriate sample space for this experi-

ment.

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