Download Chapter 6 Probability the study of randomness Probability

Chapter 6 Probability the study of randomness
Probability- describes the pattern of chance outcomes
Chance behavior is unpredictable in the short run, but has a regular and
predictable pattern in the long run
Random- if individual outcomes are uncertain but there is nonetheless a
regular distribution of outcomes in a large number of repetitions.
Probability- is the proportion of times the outcome would occur in a
very long series of repetitions. (long-term relative frequency)
Independent trials- must not influence other trials
6.2 Probability models
Probability models
Sample Space – S – the set of all possible outcomes
Event- is any outcome or set of outcomes (event is a subset of sample
space)
Probability model – mathematical description of a random phenomenon
consisting of two parts: a sample space and a way of assigning
probabilities to the events.
Rolling 2 dice
(sum)
(Probability)
(sum)
(probability)
2
1/36
8
5/36
3
2/36
9
4/36
4
3/36
10
3/36
5
4/36
11
2/36
6
5/36
12
1/36
7
6/36
Probability of rolling a five (see notation below)
P(5)
(options 1-4; 4-1; 2-3; 3-2) P(5) = 4/36 or 1/9
Make a tree diagram
Flip a coin and roll a die
Coin
Die
Final
Coin
Die
Final
Outcome
Outcome
Outcome
outcome
outcome
outcome
1
H1
1
T1
2
H2
2
T2
3
H3
3
T3
4
H4
4
T4
5
H5
5
T5
6
H6
6
T6
H
Multiplication principle
One task – a ways
Second task – b ways
Then both a and b
a b ways
coin
die
(2)
(6)
12 possible outcomes
T
Flip 4 coins
1st flip
(2)
2nd flip
(2)
3rd flip
4th flip
(2)
(2)
16 outcomes
If we want to count only # of heads
Outcomes are (0,1,2,3,4)
heads, all heads)
(no heads, one heads, two heads, three
With replacement- when you draw, you put it back.
Without replacement – when you draw you do not put it back.
Homework
Read 330 – 340 do problems 11 – 15, 17, 18
Tuesday
Probability rules
1) Any probability is a number between 0 and 1
2) All possible outcomes together must have probability 1
3) The probability that an event does not occur is 1 minus the
probability that the event does occur.
4) If two events have not outcomes in common, the probability that
one or the other occurs is the sum of their individual probabilities.
In math terms
1.) A is an event
0 P(a)
1
2.) P(s) = 1
3.) Complement – probability the event does not occur
P(Ac) = 1 – P(A)
4.) Disjoint (nothing in common)
P(A or B) = P(A) + P(B)
{ A U B } A union B
This means it is in A or B
Empty Set – Ø
If disjoint or mutually exclusive then A
B=Ø
Venn Diagram
A
B
This is a Venn Diagram of a mutually exclusive or disjoint set.
A U Ac = S
A
Ac = Ø
Benford’s Law Page 345
Homework
Read pages 340 – 350 Do problems 19-23, 26
Wednesday
Independence and multiplication
Multiplication rule for independent events
A and B are independent
P(A and B) = P(A) P(B)
Ex 6.14 page 353
Ex 6.15 Page 354
Homework read pages 354 – 355 do problems 27-29, 31
Thursday
6.3 General Probability rules
Addition rule for disjoint events
A, B, C have nothing in common
P( one or more of A,B,C) = P(A) + P(B) + P(C)
Addition rule for unions of two events
P(A or B) = P(A) + P(B) – P(A and B)
Union means all areas
A
A&B
Ex 6.17 page 362
Deb .7
Matt .5
Together .3
Draw Venn diagram
P (A and B) = Ø
B
P(A)
P(B)
Additional problem
P(A) = .24
P(B) = .31
P(A and B) = .09
Draw picture
Homework Read 364 – 365
Problems 46 – 53
Monday
Conditional Probability
P(A B) is conditional probability
Read ex 6.18 page 366
Gives probability of one event under the condition we know the other
event
This symbol means given the information that
Do ex 6.19 pages 366 – 367
P(A and B) = P(A) P(B A)
P(B A) =
Ex 6.20 page 368
Homework read pages 369 – 371 do problems 54 – 61
Extended multiplication rules
Intersection – where all events occur
(draw Venn diagram with three circles)
P(A and B and C) = P(A)P(B A) P (C (A and B))
Tree diagram
5% go on to play at college level
1.7 % enter major league professional sports
40% have career more than 3 years
Professional sports
.017
College
.983
No professional sports
.05
Male
High School
Athletes
.95
.0001
Professional sports
.9999
No professional sports
Not College
P(Playing professional sports)
(.05)(.017) + (.95)(.0001)
.00085
+ .000095 = .000945
This means 9 out of every 10,000 high school athletes play
professionally
Bayes Rule
If A and B are any events whose probability in not 0 or 1
P(A B) =
A occurring given the information that B occurred
Two events A and B that both have positive probability are independent
if
P(A and B) = P(A)P(B)
Diagram page 376
Homework do problems 62 - 64