Download Probability

Please take out a piece of notebook paper.
For each question state if it is a permutation,
combination or counting principle AND solve!
1.
You have five new books to read. You want to take two of
them with you on vacation. In how many ways can you
choose two books to take?
2. Suppose that first, second and third-place winners of a
contest are to be selected from eight students who entered.
In how many ways can the winners be chosen?
3. Pens are available in three colors, and four tips. How many
different choices of pens you have?
4. A code is being created and contains 5 characters. The first
three are letters and the last two are digits. How many
possibilities are there for codes if letters and digits cannot
be repeated?
5. What is the difference between a permutation and
combination?
Probability
 An experiment is a process, such as rolling a dice or
tossing a coin, that gives results called outcomes.
 The sample space ,S , of an experiment is the set of all
possible outcomes
 Ex: Tossing a coin

S={H,T}
 An event is any subset of the sample space of an
experiment
Sample Space Example
 If we toss a coin three times and record the results in
order, what is the sample space?
 S={HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
 What is the event of exactly two heads?
 S={HHT, HTH, THH}
 What is the event of at least two heads?
 S={HHH, HHT, HTH, THH}
Probability
 If S is the sample space of an experiment and E an
event. The probability of E is
number of elements in E
P( E ) 
number of elements in S
 Probabilities should ALWAYS be between 0 and 1
 If P(E)=1 then it is a certain event
 If P(E)=0 then it is an impossible event
Finding Probability Example
 A coin is tossed three times. What is the probability of
getting exactly two heads?
3
P(exactly 2 heads ) 
8
 At least 2 heads?
4 1
P(at least 2 heads )  
8 2
 No heads?
1
P (no heads ) 
8
Probability Example 2
 A five card hand is drawn from a standard deck of 52
cards. What is the probability that all five cards are
spades?
choosing 5 spades C (13,5)
P(5 cards are spades) 

choosing 5 cards C (52,5)
1287
 .0005
2,598,960
Probability Example 3
 A bag containing 20 tennis balls, of which 4 are
defective. If two balls are selected at random from the
bag, what is the probability that both are defective?
choosing 2 defective
C (4,2)
P(2 defective tennis balls ) 

choosing 2
C (20,2)
6
 .032
190
Complement of an event
 The complement of an event ,E , is the set of outcomes
in the sample space that are NOT in E.
P( E ' )  1  P( E )
Example
 An bag contains 10 red balls and 15 white balls. Six are
drawn at random from the bag. What is the probability
at least one is red?
P(at least 1 is red )  1  P(none are red )
all 6 are white
 1
choosing 6
C (15,6)
5005
 1
 1
 .97
C (25,6)
177,100
Mutually Exclusive Events
 Two events with no outcomes in common are mutually
exclusive.
 Example: in drawing a card from a deck, the events E:
the card is a queen, and F: the card is an ace are
mutually exclusive
P( E  F )  P( E )  P( F )
Mutually Exclusive Example
 A card is drawn from a standard deck. What is the
probability that the card is either a seven or a face
card?
 Two events:
 E: the card is a 7
 F: the card is a face card
P(7 or face card )  P(a 7)  P(a face card )
4 12 16


 .3077
52 52 52
Probability of the Union of 2 events
 If two events are not mutually exclusive then they
share outcomes
P( E  F )  P( E )  P( F )  P( E  F )
Example
 What is the probability that a card drawn from a
standard 52 card deck is either a face card or a spade?
 Events:
 E: the card is a face card
 F: the card is a spade
P( E  F )  P( E )  P( F )  P( E  F )
P(face card )  P(spade)  P(face card and spade)
12 13 3
 
 .4321
52 52 52
Probability of Independent Events
 This is used when we want to find the probability of an
event AND another event.
 Independent means the probability of the first event
does not effect the probability of the second event

Ex: tossing a fair coin
P( E  F )  P( E ) P( F )
Example
 A jar contains 5 pink marbles and 4 green marbles. A
marble is drawn at random from the jar and replaced,
then another marble is drawn. What is the probability
both marbles are pink?
P(pink and pink )  P(1st is pink )  P(2nd is pink )
5 5 25
  
 .31
9 9 81
Standard Deck of Cards
 4 suits (13 of each)
 Diamonds (Red)
 Hearts (Red)
 Spades (Black)
 Clubs (Black)
 Face Cards (one of each in every suit)
 King
 Queen
 Jack