• Part 3: Distribution and Limits
Course Format
• Recommended: A First Course in Probability, Sheldon Ross, 9th Ed.
• It is okay to use a di↵erent edition. 8th edition viewable at
https://hdl.handle.net/2027/uc1.31822037229853
https://open.umn.edu/opentextbooks/textbooks/21
http://probabilitybook.net
• Probability and Statistics, DeGroot & Schervish.
Email Policy: psdey@illinois.edu
• [Math 461] Your subject here.
• Will respond to all emails that follow those two rules.
• Preference for short, meaningful subjects lines.
• May be asked to re-send or attend oce hours.
Grading
Participation 5% Homework 15% Quiz 15% Two Midterms 30% Final 35%
Tentative grading scale:
Homework and Quiz
• Every (non-exam) week. (Excluding this week.)
• Detailed instructions for completion and submission can be found in compass2g.
• 20 points each.
• Questions will be similar to example, homework, and practice problems.
• Details coming soon...
• Final Exam: TBA
Tips for Success
• Follow the lectures!
• Don’t Quit!
• Practice!
Questions that Partha will repeatedly ask all semester to make sure that you know the answer at the end of the course otherwise he’ll think he didn’t do his job:
What is a .....
Why Study Probability? • Quantifying Uncertainty:
• usually modeled as randomness. • It appears in call centers, electronic circuits, quantum
mechanics, medical treatment, epidemics, financial investments, insurance, games (both sports and gambling), online search engines, for starters.
• Probability is a good way of quantifying and discussing what we know about uncertain things, and
• making decisions or estimating outcomes.
• Analyze Complex Systems. • Some things are too complex to be analyzed exactly (like
weather, the brain, social science) • Probability is a useful way of reducing the complexity and
providing approximations.
Two Examples
Example: How many two letter combinations are there? (There are 26 letters.)
Example: Same question as above but the same letters are not allowed.
A- A , AB , AC , - - - .
B A , B B, - - - - -
AAIAD , AC . .
BA, BIT, -
. Z1
F
=
=
5 vowels anything 26 5 Which letter
.
Examples
Example 2c: How many di↵erent 7-place license plates are possible if the first 3 places are to be occupied by letters and the final 4 by numbers?
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n ! = 1×2×3 x - i - X n
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letters numbers
26 ! 10 ! = -
23 ! 6 !
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Example: Number of subsets
Consider a set S of n elements: S = {1, 2, . . . , n}. How many di↵erent subsets of S are there? (Including the empty set ;.)
For example, for n = 2, there are 4 possible subsets: ;, {1, 2}, {1} and {2}. -
5=91,2 } , 0 , { i ) , { 23 , {1,2 } → 4 = 2
"
Consecutive hunters ! !
• Permutations
• Combinations
Permutations Consider n distinct objects; how many di↵erent ordered arrangements, called permutations, of these objects are? For example, for n = 2 objects a, b, there are two arrangements (permutations): ab, ba.
Notation: n!. Name: n factorial.
#
a. b → ab , b a
÷ ÷ -
i . ÷
Example 3c: Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many di↵erent arrangements are possible?
different
a-
Arrange the book for each subject :
iii. ii M ,
C , L , H
4 ! x 4 ! X 3 ! X l ! X 2 ! =
Combinations: Consider n distinct objects; how many di↵erent groups, called combinations, of size r (1 r n) of these objects can be formed? For example, for n = 3 objects a, b, c and r = 2, there are three possible combinations: ab, bc , ac .
Notation: n r
n 0
, be , ac
"
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i = (2) = In
Example 4b: From a group of 5 women and 7 men, how many di↵erent committees consisting of 2 women and 3 men can be formed? What if 2 of the men are feuding and refuse to serve on the committee together?
Are . ( Elf }) = ÷÷×s÷. .
(Eek ! Fresh - feuding men appearing together .
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