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Sample space
•
the collection of all
possible outcomes of a
chance experiment
–Roll a die
S={1,2,3,4,5,6}
Event
• any collection of outcomes
from the sample space
–Rolling a prime # E= {2,3,5}
Complement
• Consists of all outcomes
that are not in the event
–Not rolling an even #
EC={1,3,5}
Union
• the event A or B happening
• consists of all outcomes that
are in at least one of the two
events
– Rolling a prime # or even number
E={2,3,4,5,6}
E  AB
Intersection
• the event A and B happening
• consists of all outcomes that
are in both events
– Drawing a red card and a “2” E = {2
hearts, 2 diamonds}
E  AB
Mutually Exclusive
(disjoint)
• two events have no
outcomes in common
–Roll a “2” and a “5”
Probability
• Denoted by P(Event)
favorable outcomes
P( E ) 
total outcomes
This method for calculating probabilities is only appropriate
when the outcomes of the sample space are equally likely.
Experimental Probability
• The relative frequency at which a
chance experiment occurs
– Flip a fair coin 30 times & get 17
heads
17
30
Law of Large Numbers
• As the number of repetitions of a
chance experiment increase, the
difference between the relative
frequency of occurrence for an
event and the true probability
approaches zero.
Basic Rules of Probability
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
Complement of an event…..
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Rule 4. 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)
P( A  B) ?
P( A  B)
Yes
P( A)  P( B)
What does
this mean?
Independent?
Ex. 1) A certain brand of light bulbs are
defective five percent of the time. You
randomly pick a package of two such
bulbs off the shelf of a store. What is the
probability that both bulbs are defective?
Can you assume they are independent?
P(D & D)  .05  .05  .0025
P( A  B) ?
P( A  B)
Yes
P( A)  P( B)
Independent?
No
P( A)  P( B | A)
Ex. 2) There are seven girls and eight boys
in a math class. The teacher selects two
students at random to answer questions on
the board. What is the probability that
both students are girls?
Are these events independent?
7 6
P(G & G) 

 .2
15 14
Ex 6) 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)
P(Spade & Spade) = 1/4 · 12/51 = 1/17
The probability of getting a spade given
that a spade has already been drawn.
Rule 5. Addition
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)
P( E  F ) ?
P( E  F )
Yes
P( E )  P( F )
What does
this mean?
Mutually exclusive?
Ex 3) A large auto center sells cars made by
many different manufacturers. Three of these
are Honda, Nissan, and Toyota. (Note: these
are not simple events since there are many
types of each brand.) 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
P( E  F ) ?
P( E  F )
Yes
P( E )  P( F )
Mutually exclusive?
No
P( E )  P( F )  P( E  F )
Ex. 4) Musical styles other than rock
and pop are becoming more popular. A
survey of college students finds that the
probability they like country music is
.40. The probability that they liked jazz
is .30 and that they liked both is .10.
What is the probability that they like
country or jazz?
P(C or J) = .4 + .3 -.1 = .6
P( E  F ) ?
P( E  F )
Mutually exclusive?
No
Yes
P( E )  P( F )
P( E )  P( F )  P( E  F )
Independent?
Yes
P( E)  P( F )
Ex 5)
If P(A) = 0.45, P(B) = 0.35, and A &
B are independent, find P(A or B).
Is A & B disjoint?
NO, independent events cannot be disjoint
If A &
are –disjoint,
are they
P(A or B) = P(A)
+B
P(B)
P(A & B)
Disjoint
events are
independent?
Disjoint events
doIf not happen at the same
dependent!
How can you
independent,
time.
find the
P(A or B) =So,
.45if +A multiply
.35
.45(.35)
=
0.6425
occurs, can B occur?
probability of
A & B?
If a coin is flipped & a die rolled at the same time, what is the
probability that you will get a tail or a number less than 3?
Sample Space
H1
Coin is TAILS
H2
H3
H4
Roll is LESS THAN 3
H5
T6
T4
T2
H1
H4
T3
T1
H2
T5
H6
H3
H5
H6
T1 T2
T3
T4
T5
T6
Flipping a coin and rolling a
die are independent events.
Independence also implies the
events are NOT disjoint
(hence the overlap).
Count T1 and T2 only once!
P (tails or even) = P(tails) + P(less than 3) – P(tails & less than 3)
= 1/2 + 1/3 – 1/6
= 2/3
Ex. 7) A certain brand of light bulbs are
defective five percent of the time. You
randomly pick a package of two such bulbs
off the shelf of a store. What is the probability
that exactly one bulb is defective?
P(exactly one) = P(D & DC) or P(DC & D)
= (.05)(.95) + (.95)(.05)
= .095
Ex. 8) A certain brand of light bulbs are
defective five percent of the time. You randomly
pick a package of two such bulbs off the shelf of
a store. What is the probability that at least one
bulb is defective?
P(at least one) = P(D & DC) or P(DC & D) or (D & D)
= (.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. 8 revisited) A certain brand of light
bulbs are defective five percent of the
time. You randomly pick a package of two
such bulbs off the shelf of a store.
What is the probability that at least one
bulb is defective?
P(at least one) = 1 - P(DC & DC)
.0975
Ex 9) 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
Rule 7: Conditional Probability
• A probability that takes into
account a given condition
P(A  B)
P(B | A) 
P(A)
P(and)
P(B | A) 
P(given)
Ex 10) In a recent study it was
found that the probability that a
randomly selected student is a
girl is .51 and is a girl and plays
sports is .10. If the student is
female, what is the probability
that she plays sports?
P(S  F) .1
P(S | F) 

 .1961
P(F)
.51
Ex 11) The probability that a
randomly selected student plays
sports if they are male is .31. What
is the probability that the student is
male and plays sports if the
probability that they are male
is .49?
P(S  M)
x
P(S | M) 
.31 
P(M)
.49
x  .1519