Probability I

1
Probability I
2
Notes

• Probability of A occurring
• P(A)
• Sum of all possible outcomes 1

3
Sample Space

• the collection of all possible outcomes of a
  chance experiment
• Roll a die S1,2,3,4,5,6

4
Relative Frequency

• Of Occurrences of Event
• Trials
• Not rolling a even
• EC1,3,5

5
The Law of Large Numbers

• The long run relative frequency will approach the
  actual probability as the number of trails
  increases
• Coins? 2, 10, 20.

6
Event

• any collection of outcomes from the sample space
• Rolling a prime E 2,3,5

7
Complement

• Consists of all outcomes that are not in the
  event
• Not rolling a even
• EC1,3,5
• P(A) 1 P(A)

8
Mutually Exclusive (disjoint)

• two events have no outcomes in common
• Roll a 2 or a 5
• Draw a Black card or a Diamond

9
Not -Mutually Exclusive (Non- disjoint)

• two events have outcomes in common
• Draw a Black card or a Spade

10
UnionDisjoint

• the event A or B happening
• consists of all outcomes that are in at least one
  of the two events
• Draw a Black card or a Diamond

11
UnionDisjoint

• Draw a Black card or a Diamond
• P(B U D) P(B) P(D)

12
Intersection

• the event A and B happening
• consists of all outcomes that are in both events
• Draw a Black card and a 7

13
Intersection

• P(B S) P(B)P(S)
• Draw a Black card and a 7

U
14
UnionNot Disjoint

• the event A or B happening BUT WE CANT Double
  Count!
• Draw a Black card or a 7
• P(B or 7) P(B) P(7) P(B and 7)

15
Venn Diagrams

• Used to display relationships between events
• Helpful in calculating probabilities

16
Venn Diagram Mutually Exclusive / Disjoint
events
A B
17
Venn Diagram Not Mutually Exclusive / Non-
Disjoint events
A B
18
Venn diagram - Complement of A
A
19
Venn diagram - A and B
A B
20
Stat
Cal
Com Sci
Statistics Computer Science not Calculus
21
Stat
Cal
Stat
Cal
Com Sci
Com Sci
(Statistics or Computer Science) and not Calculus
22
Two- Way Table

1. P ( has pierced ears. )
2. P( is a male or has pierced ears. )
3. P( is a female or has pierced ears )

23
(No Transcript)
24
Basic Rules of Probability

• Rule 1. Legitimate Values
• For any event E,
• 0 lt P(E) lt 1
• Rule 2. Sample space
• If S is the sample space,
• P(S) 1

25
Rule 3. Complement For any event E, P(E)
P(not E) 1 Or P(not E) 1 P(E)
26
Rule 4. Addition (A or B) If two events E F
are disjoint, P(E or F) P(E)
P(F) (General) If two events E F are not
disjoint, P(E or F) P(E) P(F) P(E F)
27
Ex 1) A large auto center sells cars made by many
different manufacturers. Three of these are
Honda, Nissan, and Toyota. Suppose that P(H)
.25, P(N) .18, P(T) .14.
Are these disjoint events?
yes
P(H or N or T)
.25 .18 .14 .57
P(not (H or N or T)
1 - .57 .43
28
Independent

• Two events are independent if knowing that one
  will occur (or has occurred) does not change the
  probability that the other occurs
• Flip a Coin and Get Heads. Flip a coin again.
  P(T)
• Draw a 7 from a deck. Draw another card. P(8)

Independent
Not independent
29
Rule 5. Multiplication If two events A B are
independent, General rule
30
The probability that a student will receive a
state grant is 1/3, while the probability she
will be awarded a federal grant is ½. If
whether or not she receives one grant is not
influenced by whether or not she receives the
other, what is the probability of her receiving
both grants?
31
Suppose a reputed psychic in an extrasensory
perception (ESP) experiment has called heads or
tails correctly on TEN successive coin flips.
What is the probability that her guessing would
have yielded this perfect score?
32

• Tree Diagrams

Consider flipping a coin twice. What is the
probability of getting two heads?
Sample Space HH HT TH TT
33

• Tree Diagrams

• Getting Tails Twice

34

• Example Teens with Online Profiles
• The Pew Internet and American Life Project finds
  that 93 of teenagers (ages 12 to 17) use the
  Internet, and that 55 of online teens have
  posted a profile on a social-networking site.
• What percent of teens are online and have posted
  a profile?

51.15 of teens are online and have posted a
profile.
35
Ex. 3) A certain brand of cookies are stale 5 of
the time. You randomly pick a package of two such
cookies off the shelf of a store. What is the
probability that both cookies are stale? Can you
assume they are independent?
36
Ex 5) Suppose I will pick two cards from a
standard deck without replacement. What is the
probability that I select two spades?
Are the cards independent?
NO
P(A B) P(A) ? P(BA)
Read probability of B given that A occurs
P(Spade Spade) 1/4 ? 12/51 1/17
The probability of getting a spade given that a
spade has already been drawn.
37

• Ex. 6) A certain brand of cookies are stale 5 of
  the time. You randomly pick a package of two such
  cookies off the shelf of a store. What is the
  probability that exactly one cookie is stale?

P(exactly one) P(S SC) or P(SC S)
(.05)(.95) (.95)(.05)
.095
38
Ex. 7) A certain brand of cookies are stale 5 of
the time. You randomly pick a package of two such
cookies off the shelf of a store. What is the
probability that at least one cookie is stale?
P(at least one) P(S SC) or P(SC S) or (S
S) (.05)(.95) (.95)(.05)
(.05)(.05) .0975
39
Rule 6. At least one The probability that at
least one outcome happens is 1 minus the
probability that no outcomes happen. P(at least
1) 1 P(none)
40
Ex. 7 revisited) A certain brand of cookies are
stale 5 of the time. You randomly pick a package
of two such cookies off the shelf of a store.
What is the probability that at least cookie is
stale?
P(at least one)
1 P(SC SC)
.0975
41
Ex 8) For a sales promotion the manufacturer
places winning symbols under the caps of 10 of
all Dr. Pepper bottles. You buy a six-pack.
What is the probability that you win something?
P(at least one winning symbol) 1 P(no
winning symbols)
1 - .96 .4686
42
Warm Up
Allergies
Female Male Total
Allergies 10 8 18
No Allergies 13 9 22
Total 23 17 40

1. What is the probability of not having allergies?
2. What is the probability of having allergies if
   you are a male?
3. Are the events Female and allergies
   independent? Justify your answer.

43
Conditional Probability and Independence
Handedness Female Male Total
Left 3 1 __
Right 18 8 __
Total __ __ __

1. Are the events female and right handed
   independent?

44
Rule 7 Conditional Probability

• A probability that takes into account a given
  condition

45

• .
• What is the probability that a randomly selected
  resident who reads USA Today also reads the New
  York Times?

There is a 12.5 chance that a randomly selected
resident who reads USA Today also reads the New
York Times.
46

• Using Table D

• When performing a random simulation we can use
  Table D.
• Lets say I have a 30 Chance of winning a class
  lottery.

47
Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
What is the probability that the driver is a
student?
48
Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
What is the probability that the driver is staff
and drives an Asian car?
49
Probabilities from two way tables
Stu Staff Total American 107 105 212 Eu
ropean 33 12 45 Asian 55 47 102 Total 195 16
4 359
If the driver is a student, what is the
probability that they drive an American car?
Condition
50
Whiteboard Challenge
51
The probability of any outcome of a random
phenomenon is (a) the precise degree of
randomness present in the phenomenon. (b) any
number as long as it is greater than 0 and less
than 1. (c) either 0 or 1, depending on whether
or not the phenomenon can actually occur or
not. (d) the proportion of times the outcome
occurs in a very long series of repetitions. (e)
none of the above.
52

• A randomly selected student is asked to respond
  Yes, No, or Maybe to the question Do you intend
  to vote in the next presidential election? The
  sample space is Yes, No, Maybe . Which of the
  following represents a legitimate assignment of
  probabilities for this sample space?
• 0.4, 0.4, 0.2
• 0.4, 0.6, 0.4
• 0.3, 0.3, 0.3
• 0.5, 0.3, 0.2
• 1/4, 1/4, 1/4

53
You play tennis regularly with a friend, and from
past experience, you believe that the outcome of
each match is independent. For any given match
you have a probability of 0.6 of winning. The
probability that you win the next two matches
is (a) 0.16. (b) 0.36. (c) 0.4. (d) 0.6. (e) 1.2.
54
There are 10 red marbles and 8 green marbles in a
jar. If you take three marbles from the jar
(without replacement), the probability that they
are all red is (a) 0.069(b) 0.088 (c) 0.147
(d) 0.171 (e) 0.444
55
Jolor and Mi Sun are applying for summer jobs at
a local restaurant. After interviewing them, the
restaurant owner says, The probability that I
hire Jolor is 0.7, and the probability that I
hire Mi Sun is 0.4. The probability that I hire
at least one of you is 0.9. What is the
probability that both Jolor and Mi Sun get hired?
(a) 0.1 (b) 0.2 (c) 0.28 (d) 0.3 (e) 1.1
56
Select a random integer from 100 to 100. Which
of the following pairs of events are mutually
exclusive (disjoint)? (a) A the number is odd
B the number is 5(b) A the number is even B
the number is greater than 10 (c) A the number
is less than 5 B the number is negative. (d)
A the number is above 50 B the number is less
than 20. (e) A the number is positive B the
number is odd.
57
A recent survey asked 100 randomly selected
adult Americans if they thought that women should
be allowed to go into combat situations. Here are
the results, classified by the gender of the
subject Gender Yes No
Male 32 18
Female 8 42 The probability of a
Yes answer, given that the person was Female,
is (a) 0.08 (b) 0.16 (c) 0.20 (d) 0.40 (e)
0.42
58
A recent survey asked 100 randomly selected
adult Americans if they thought that women should
be allowed to go into combat situations. Here are
the results, classified by the gender of the
subject Gender Yes No
Male 32 18
Female 8 42 _______________________
_______________________ The probability that a
randomly selected subject in the study is Male or
answered No is (a) 0.18 (b) 0.36 (c) 0.68
(d) 0.92 (e) 1.10
59
An airline estimates that the probability that a
random call to their reservation phone line
result in a reservation being made is 0.31. This
can be expressed as P(call results in
reservation) 0.31. Assume each call is
independent of other calls. Describe what the
Law of Large Numbers says in the context of this
probability.
60
An airline estimates that the probability that a
random call to their reservation phone line
result in a reservation being made is 0.31. This
can be expressed as P(call results in
reservation) 0.31. Assume each call is
independent of other calls. What is the
probability that none of the next four calls
results in a reservation?
61
An airline estimates that the probability that a
random call to their reservation phone line
result in a reservation being made is 0.31. This
can be expressed as P(call results in
reservation) 0.31. Assume each call is
independent of other calls. You want to
estimate the probability that exactly one of the
next four calls result in a reservation being
made. Describe the design of a simulation to
estimate this probability. Explain clearly how
you will use the partial table of random digits
below to carry out five simulations. 188 87370
88099 89695 87633 76987 85503 26257 51736 189
88296 95670 74932 65317 93848 43988 47597 83044
190 79485 92200 99401 54473 190 34336 82786
05457 60343 191 40830 24979 23333 37619 56227
95941 59494 86539 192 32006 76302 81221 00693
95197 75044 46596 11628
62
(No Transcript)