Transcript Probability I

 Probability of A occurring
 P(A)
 Sum of all possible outcomes = 1

the collection of all
possible outcomes of a
chance experiment
 Roll a die
S={1,2,3,4,5,6}
# OF OCCURRENCES OF EVENT
#TRIALS
 Not rolling a even #
EC={1,3,5}
The long run relative
frequency will approach
the actual probability as
the number of trails
increases
 Coins? 2, 10, 20.
any collection of outcomes
from the sample space
 Rolling a prime # E= {2,3,5}
Consists of all outcomes
that are not in the event
 Not rolling a even #
EC={1,3,5}
P(A) = 1 – P(A)
two events have no
outcomes in common
 Roll a “2” or a “5”
 Draw a Black card or a Diamond
two events have outcomes
in common
 Draw a Black card or a Spade
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
E  AB
 Draw a Black card or a Diamond
P(B U D) = P(B) + P(D)
the event A and B happening
consists of all outcomes that
are in both events
 Draw a Black card and a 7
E  AB
U
P(B S) = P(B)•P(S)
 Draw a Black card and a 7
E  AB
 the event A or B happening BUT
WE CAN’T Double Count!
 Draw a Black card or a 7
 P(B or 7) = P(B) + P(7) – P(B and 7)
Used to display relationships
between events
Helpful in calculating
probabilities
Stat
Cal
Com Sci
Statistics & Computer Science & not Calculus
Stat
Cal
Stat
Cal
Com Sci
Com Sci
(Statistics or Computer Science) and not Calculus
(a) P ( has pierced ears. )
(b) P( is a male or has pierced ears. )
(c)P( is a female or has pierced ears )
Rule 1. Legitimate Values
For any event E,
0 < P(E) < 1
Rule 2. Sample space
If S is the sample space,
P(S) = 1
Rule 3. Complement
For any event E,
P(E) + P(not E) = 1
Or
P(not E) = 1 – P(E)
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)
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?
P(H or N or T) =
yes
.25 + .18+ .14 = .57
P(not (H or N or T) =
1 - .57 = .43
 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)
Independent
 Draw a 7 from a deck. Draw another card. P(8)
Not independent
RULE 5. MULTIPLICATION
If two events A & B are
independent,
P(A & B)  P(A)  P(B)
General rule:
P(A & B)  P(A)  P(B | A)
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?
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?
TREE DIAGRAMS
Consider flipping a
coin twice.
What is the
probability of
getting two heads?
Sample Space:
HH HT TH TT
TREE DIAGRAMS
 Getting Tails Twice
 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?
P(online) = 0.93
P(profile | online) = 0.55
P(online and have profile) = P(online)× P(profile | online)
= (0.93)(0.55)
= 0.5115
51.15% of teens are online and have
posted a profile.
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?
P(D & D)  .05  .05  .0025
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(B|A)
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.
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
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
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)
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
Ex 8) For a sales promotion the
manufacturer places winning
symbols under the caps of 10% of all
Dr. Pepper bottles. You buy a sixpack. What is the probability that
you win something?
P(at least one winning symbol) =
1 – P(no winning symbols)
1 - .96 = .4686
WARM UPAllergies
Allergies
Female
10
Male
Total
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.
Handedness
Female
Male
Total
3
1
__
Right
18
8
__
Total
__
__
__
Left
1. Are the events “female” and “right handed”
independent?
A probability that takes
into account a given
condition
P(A  B)
P(B | A) 
P(A)
P(and)
P(B | A) 
P(given)
.
What is the probability that a randomly
selected resident who reads USA Today also
reads the New York Times?
P(A Ç B)
P(B | A) =
P(A)
P(A Ç B) = 0.05
P(A) = 0.40
0.05
P(B | A) =
= 0.125
0.40
There is a 12.5% chance that a randomly selected resident who
reads USA Today also reads the New York Times.
 When performing a random simulation we
can use Table D.
 Lets say I have a 30% Chance of winning a
class lottery.
American
European
Asian
Total
Stu
107
33
55
195
Staff
105
12
47
164
Total
212
45
102
359
What is the probability that the driver is a student?
195
P (Student ) 
359
American
European
Asian
Total
Stu
107
33
55
195
Staff
105
12
47
164
Total
212
45
102
359
What is the probability that the driver is staff and
drives an Asian car?
47
P (Staff and Asian ) 
359
American
European
Asian
Total
Stu
107
33
55
195
Staff
105
12
47
164
Total
212
45
102
359
If the driver is a student, what is the probability
that they drive an American car?
107
P (American |Student ) 
195
Condition
Whiteboard Challenge
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.
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?
(a)0.4, 0.4, 0.2
(b) 0.4, 0.6, 0.4
(c) 0.3, 0.3, 0.3
(d) 0.5, 0.3, –0.2
(e) 1⁄4, 1⁄4, 1⁄4
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.
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
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
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.
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
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
Male
Female
Yes
32
8
No
18
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
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.
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?
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