Download Chapter 4 Elementary Probability Theory

Chapter 4 Elementary
Probability Theory
What is Probability?
 Probability is a numerical measure between 0 and 1 that
describes the likelihood that an event will occur.
Probabilities closer to 1 indicate that the event is more
likely to occur. Probabilities closer to 0 indicated that
the event is less likely to occur.
Note:
 P(A) = probability of event A; you read it as “P of A”.
 P(A)=1, the event A is certain to occur
 P(A)=0, the event A is certain to not occur
 Binary number works like this…1 means it’s true, 0 means
false.
See if you understand this
statement:
 “There are only 10 types of people in the world: those
who understand binary, and those who don't”
Anyways…Probability
Assignments
 1) A probability assignment based on intuition
incorporates past experience, judgment, or opinion to
estimate the likelihood of an event
 2) A probability assignment based on relative frequency
uses the formula
 Probability of event=relative frequency=
𝑓
𝑛
 Where f is the frequency of the event occurrence in a
sample of n observations
 3) A probability assignment based on equally likely
outcomes uses the formula
 Probability of event=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑒𝑣𝑒𝑛𝑡
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Examples
 Intuition – NBA announcer claims that Kobe makes 84%
of his free throws. Based on this, he will have a high
chance of making his next free throw.
 Relative frequency – Auto Fix claims that the probability
of Toyota breaking down is .10 based on a sample of
500 Toyota of which 50 broke down.
 Equally likely outcome - You figure that if you guess on a
SAT test, the probability of getting it right is .20
Group Work
 Create a situation for each of the probability
assignments. (intuition, relative frequency, equally
likely outcome)
 Show me
Law of Large Numbers
 In the long run, as the sample size increases, the
relative frequencies of outcomes get closer to the
theoretical (or actual) probability value
 Example: The more numbers you ask, the more
likelihood that P(getting a girl’s real number)=1
Law of Large Numbers examples:
 The more numbers you ask, the more likelihood that
P(getting a (hot) girl’s real number)=1
 Then after collecting all the numbers, the more girls you
ask out on a date, the more likelihood that P(getting a
date)=1
Some other real life examples:
 Casino (the more you play, the more you lose)
 Insurance (the more people you insure, the less the
likelihood the company have to pay for the insurance
benefits)
Statistical Experiment
 Statistical experiment or statistical observation can be
thought of as any random activity that results in a
definite outcome
 An event is a collection of one or more outcomes of a
statistical experiment or observation
 Simple event is one particular outcome of a statistical
experiment
 The set of all simple events constitutes the sample space
of an experiment
Example: Blue eyes vs Brown eyes
(relating to biology)
 Brown eyes’ genotype is Bb or BB
 Blue eyes’ genotype is bb
 If your Dad has Brown eyes (and his dad has blue eyes)
and your Mom has blue eyes, what’s the probability that
you have blue eyes?
Answer (using sample space)
Dad
Mom
B
b
b
Bb
bb
b
Bb
bb
P(blue eyes)=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
2
1
=4=2
Group Work (use sample space):
 You are running out of time in a true/false quiz. You only
have 4 questions left! How should you guess?
 P(all false)=
P(3 false)=
 P(all true)=
P(2 false)=
 P(1 true)=
P(1 false)=
 P(2 true)=
 P(3 true)=
Answer

Your sample space should have 16 different combinations
TTTT
FTTT
TFTT
TTFT
TTTF
FFTT
FTFT
FTTF
TFFT
TFTF
TTFF
FFFT
TFFF
FTFF
FFTF
FFFF

P(all false)= 1/16
P(3 false)= 4/16

P(all true)= 1/16
P(2 false)= 6/16

P(1 true)=
P(1 false)= 4/16

P(2 true)= 6/16

P(3 true)=

You will probably choose 2 true and 2 false
4/16
4/16
Note:
 The sum of the probabilities of all simple events in a
sample space must equal 1
 The complement of event A is the event that A does not
occur. 𝐴𝑐 designates the complement of event A.
Furthermore,
 1) P(A)+P(𝐴𝑐 )= 1
 P(event A does not occur)=P 𝐴𝑐 = 1 − 𝑃(𝐴)
Example:
 P(getting A in Mr. Liu’s class)+P(not getting A in Mr. Liu’s
class) =1
 P(getting A in Mr. Liu’s class)=.15
 What’s the P(not getting A in Mr. Liu’s class)?
Answer
 P(not getting A in Mr. Liu’s class)= .85
Group Work
 P(having a date on a Friday)=1/7
 What’s the P(not having a date on a Friday)?
Answer
 6/7
Homework Practice:
 Pg 130 #1-6 (all), 7-13 (odd)
Compound Events
Consider these two situation
 P(5 on 1st die and 5 on 2nd die)
 P(ace on 1st card and ace on 2nd card)
 What is the difference between these two situation?
The answer
 In the first situation, the first result does not effect the
outcome of the 2nd result.
 In the second situation, the first result does effect the
outcome of the 2nd result.
Independent
 Two events are independent if the occurrence or
nonoccurrence of one does not change the probability
that the other will occur
 What does it mean if two events are dependent?
Multiplication rule for independent
events
 P(A and B)=𝑃 𝐴 ∗ 𝑃(𝐵)
 This means event A AND event B both have to happen!!!
You multiply the events. You find the probability of two
events happening together.
 This is the formula if event A and event B are
independent.
What if the events are
dependent?
 Then we must take into account the changes in the
probability of one event caused by the occurrence of the
other event.
Sample
Space
A and B
A
B
General multiplication rule for any
events
 P(A and B)=𝑃 𝐴 ∗ 𝑃 𝐵 𝐴
 Or
 P(A and B)=𝑃 𝐵 ∗ 𝑃 𝐴 𝐵
What is P 𝐴 𝐵 ?
 It is known as conditional probability
 𝑃 𝐴𝐵 =
𝑃 𝐴 𝑎𝑛𝑑 𝐵
𝑃 𝐵
 𝑃 𝐴𝐵 =
𝑃
𝐵𝐴
= “Probability of event A given event B”
𝑃(𝐴)
𝑃(𝐵)
 Quick group work:
 What is P 𝐵 𝐴 ?
Conditional Probability Example:
 Your friend has 2 children. You learned that she has a
boy named Rick. What is the probability that Rick’s
sibling is a boy?
 Take a guess 
Answer
 If you guessed ½ or 50%, that is incorrect.
 First: Think about all the possible outcomes
 S {BB, BG, GB, GG}
 What is P(boy and boy)?
 What is P(boy)?
 You want to find 𝑃 𝐵 𝐵
𝑃 𝑏𝑜𝑦 𝑎𝑛𝑑 𝑏𝑜𝑦
=
𝑃 𝑏𝑜𝑦
=
1
4
3
4
1
=3
Group Work
 A machine produce parts that’s either good (90%),
slightly defective (2%) or obliviously broken (8%). The
parts gets through an automatic inspection machine that
is able to find the oblivious broken parts and throw them
away. What is the probability of the quality part that
make it through and get shipped?
Answer
 P(Good given not broken)=
.978 = 97.8%
𝑃 𝐺𝑜𝑜𝑑 𝑎𝑛𝑑 𝑛𝑜𝑡 𝑏𝑟𝑜𝑘𝑒𝑛
𝑃(𝑛𝑜𝑡 𝑏𝑟𝑜𝑘𝑒𝑛)
.90
= .92 =
Relax!
 Conditional Probability can be very intriguing and
complicated. We won’t go into any more in depth…..or
maybe….
Note:
Very important to understand about probability is that are
the events dependent or independent.
Group Work
 Suppose you are going to throw 2 fair dice. What is the
probability of getting a 3 on each die?
 A) Is this situation independent or dependent?
 B) Create all the sample space (all the potential
outcomes)
 C) What is the probability?
Answer
 A) Independent because one event does not affect the
second event
 B) You should have 36 total outcomes
 C) 1/36
Group Work
 I took a die away. Now you only have ONE die! Again
you toss the die twice. What is the probability of getting
a 1 on the first and 4 on the second try?
Answer
 It is still an independent event!
 1/36
Note:
 The last two examples are considered multiplication rule,
independent events.
Group Work
 Mr. Liu has a 80% probability of teaching statistics next
year. Mr. Riley has a 15% probability of teaching
statistics next year. What is the probability that both Mr.
Liu and Mr. Riley teach statistics next year?
Answer
 .8*.15=.12 or 12% probability
Now comes the dependent events
 Suppose you have 100 Iphones. The defective rate of
iphone is 10%. What is the probability that you choose
two iphones and both are defective?
Answer
 P(1st defective camera)=10/100
 P(2nd defective camera)=9/99
 P(1st defective camera and 2nd defective camera)=
1
11
1
= 110 = .009%
1
10
∗
Group work
 What is the probability of getting tail and getting a 3 on
a die and getting an ace in a deck of cards?
Answer
 P(tail)=1/2
 P(3)=1/6
 P(ace)=4/52=1/13

1
2
1
1
1
∗ 6 ∗ 13 = 156
Addition Rules
 You use addition when you want to consider the
possibility of one event OR another occurring
Example:
4
4
8
2
 P(Jack or King)=P(Jack)+P(King)=52 + 52 = 52 = 13
Group Work: And or Or?
 1) Satisfying the humanities requirement by taking a
course in the history of Japan or by taking a course in
classical literature
 2) Buying new tires and aligning the tires
 3) Getting an A in math but also in biology
 4) Having at least one of these pets: cat, dog, bird,
rabbit
Answer
 1) or
 2) and
 3) and
 4) or
Note:
 Two events are mutually exclusive or disjoint if they
cannot occur together. In particular, events A and B are
mutually exclusive if P(A and B)=0
Addition rule for mutually
exclusive events A and B
 P(A or B)=P(A)+P(B)
 General Rule for any events A and B
 P(A or B)=P(A)+P(B)-P(A and B)
 Remember in mutually exclusive events P(A and B)=0
Group Example:
Employee
type
Democrat
(D)
Republican
(R)
Independent Row Total
(I)
Executive
(E)
5
34
9
48
Production
Worker (PW)
63
21
8
92
Column Total 68
55
17
140 grand
total
a)
b)
c)
d)
e)
Compute P(D) and P(E)
Compute 𝑃 𝐷 𝐸
Are events D and E independent?
Compute P(D and E)
Compute P(D or E)
Answer
 A) P(D)=68/140
P(E)=48/140
5
 B) 𝑃 𝐷 𝐸 = 48
 C) Determine if P(D)=𝑃 𝐷 𝐸 , they are not the same! So
they are not independent
 D) P(D and E)=5/140
 E) P(D or E)=
68
48
5
+
−
140
140
140
=
111
140
Homework Practice:
 Pg 146 #1,2,5,7,9,10,14,19
Conditional Probability extension
 Bayes’s theorem: It uses conditional probabilities to
adjust calculations so that we can accommodate new
relevant information.
 The special case where event B is partitioned into only
two mutually exclusive events.
Formula
 𝑃 𝐴𝐵 =
𝑃
𝑃
𝐵𝐴
𝐵𝐴
𝑃 𝐴 +𝑃
𝑃(𝐴)
𝐵 𝐴𝑐
 𝐴𝑐 = 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝐴
𝑃(𝐴𝑐 )
Example (Real Life Extension):
How accurate is “one” pregnancy
test?
 Supposedly pregnancy test strip claims it is 99%
accurate.
 false-positive and false-negative
NOT Pregnant
Pregnant!!!!
Pregnancy
Test Negative
True Negative
False Negative
Pregnancy
Test Positive
False Positive
True
Positive!!
Procedure
 So you want to know 𝑃 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑇𝑒𝑠𝑡 =
𝑃 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 𝑃(𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡)
𝑃 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 𝑃 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 +𝑃 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑁𝑜𝑡 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡
𝑃(𝑛𝑜𝑡 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡)
 𝑃 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 = .99
 𝑃 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 = .15 (𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑟𝑒𝑎𝑠𝑒𝑎𝑟𝑐ℎ, 𝑖𝑡 ′ 𝑠 𝑖𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 15% − 25%)
 𝑃 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑁𝑜𝑡 𝑃𝑟𝑒𝑔𝑛𝑎𝑛𝑡 =.50
 𝑃 𝑁𝑜𝑡 𝑝𝑟𝑒𝑔𝑛𝑎𝑛𝑡 = .85

.99(.15)
.99 .15 +.50(.85)
=
.1485
.1485+.425
=
.1485
.5735
= .26 so 26%
 But if you do the test more often, then the accuracy of the test
increases.
Trees and Counting Techniques
Tree diagram:
 A tree diagram shows all the possible outcomes of an
event.
 All possible outcomes of an event are shown by a tree
diagram.
Example using tree diagrams:
 If a coin and a dice are tossed simultaneously, what is
the probability of getting tail and even number?
Answer
 1/4
Group Work using tree diagram:
 You are on a sports team. What is the probability that
out of three games, you win two of them?
Tree Diagram with Probability:
 You have 7 balls, 4 are blue and 3 are green. What is
the probability that when you pick the balls, you get
green on 1st and blue one 2nd?
Shown in class
Group Work:
 You make free throws 85% of the time. What is the
probability of making at least one out of the three?
 P(make 1 out of 3)=99.66% of the time
Factorials
 ! Means factorial
 0!=1
 1!=1
 n!=(n)(n-1)(n-2)(n-3)….
What is 6! ?
 6!=6*5*4*3*2*1=720
Combination vs Permutation
 Combination: 𝐶𝑛,𝑟 𝑜𝑟 𝐶
 Permutation: 𝑃𝑛,𝑟 𝑜𝑟 𝑃
𝑛
𝑜𝑟 𝑛𝐶𝑟 = 𝑟!
𝑟
𝑛
𝑜𝑟 𝑛𝑃𝑟 =
𝑟
𝑛!
𝑛−𝑟 !
𝑛!
𝑛−𝑟 !
Combination:
 Order does not matter! It is not important
 If you have 3,1,2, it is the same as 1,3,2 because they all
have 1,2,3
 Different Arrangement of things
 Combination is choosing
Permutation:
 Order does matter! It is important
 Note: Permutation is position
Combination vs Permutation
continue:
 Both of them break down into two different category:
 Combination with repetition (ex: ice cream scoops)
 Combination without repetition (ex: lottery)
 Permutation with repetition (ex: lock in locker room)
 Permutation without repetition (ex: marathon race)
Reading activity:
 http://www.mathsisfun.com/combinatorics/combinations
-permutations.html
 Read the different examples
Whew that was a lot of
reading…now do some examples.
 You have 8 people, what are the number of possible
ordered seating arrangement for 5 chairs
Answer
 Permutation, 𝑃8,5 =
8!
8−5 !
= 6720
Group Work:
Gamestop has 25 new games this month and you decided
to buy 5 of them. How many different arrange of game you
can have?
Answer
25!
 Combination=𝐶25,5 = 5!20! = 53130
Group Work: MEGA Million
 What is the chance of winning the jackpot for MEGA
Million?
 You have 5 slots + 1 slot for MEGA number
 The first 5 slots are numbers between 1-75, mega
number is number between 1-15
Answer

75!
∗
70!5!
15 = 258890850
 .00000000386
 Or .000000386%
 1 in 258890850
Group Work: Powerball lottery
 What is the chance of winning the Jackpot for Powerball?
 You have 5 slots+1 slot for Power number
 The first 5 slots are numbers between 1-56, powerball
slot is number between 1-35
Answer
 1 in 175,223,510
Interesting fact
 Even though it’s harder to win the Jackpot, for overall
winning chance, you have more chance for MEGA million
than Powerball
Homework Practice
 Pg 160 #1-27 every other odd