Probability
Webinar Math 201
Liberty University
Dr. Steve Armstrong
[email protected]
Probability Experiments
• A probability experiment is
– An action or trial
– Specific results are obtained
• The result is an outcome
– Set of all possible outcomes is the sample space
• An event is a subset of the sample space
Coin Flip Experiment
• Watch the coin flip …
• Possible events?
• Sample space?
Roll of single die?
• Possible events?
• Sample space?
Roll of two dice?
• Possible events?
• Sample space?
Hand of Three Cards
• Possible events?
• Sample space?
Fundamental Counting Principle
• Possible events?
– Deal one card from deck of 52
– Next card dealt from deck of 51, next from 50
• Possible events = 52  51  50 = 132,600
• But wait a minute … 3, 4, 5 is the same
as 4, 5, 3
We’ll come back to this
Fundamental Counting Principle
• Consider a license plate
• Number 1234 is not the same as 4321, even
though the same digits are chosen
• Similarly BCD would be different from CBD or
DCB
• So in some cases order matters
Fundamental Counting Principle
• An ordered arrangement of objects (letters,
numbers) is called a permutation.
• If we say no digits or letters repeated then we
have
10  9  8 7  26  25  24 = 78,624,000
Permutations
• Permutation of n objects taken r at a time
n!
n Pr 
 n  r !
• For the numbers on our license plate
• For the letters
26!
 15, 600
26 P3 
 23!
10!
 5040
10 P4 
 6 !
Permutations
• Geogebra App available
Does Order Matter?
• For a hand of cards? No
• This is called a “combination”
• Combination of n objects taken r at a time
n!
n Cr 
 n  r !  r !
– Where r ≤ n
Combinations
• So, our 3 card hands combinations
would be
52!
 22,100
52 C3 
49!  3!
• We had 52  51  50 = 132,600 … but divide by
the 3! = 6 and you get the 22,100
Combinations
• I’ve created a Geogebra app for this one also
– Those factorials make big numbers
– Note the formulas come up integers
Probability
• We use combinations and permutations to
determine probabilities
• Classical (theoretical) probability
Number of outcomes in event E
P( E ) 
Total number of outcomes in sample space
?
P(rolling a 4) 
?
Roll of Two Dice?
Number of ways to get a 9
P(rolling a 9) 
Number of ways two dice can roll
Roll of Two Dice?
The Law of Large Numbers
• Experiment is repeated over and over
– empirical probability approaches the theoretical
(actual) probability of the event.
• 36 possible ways for a pair of dice to roll.
– For each possible sum, how many ways can it
happen?
– Probability for each sum?
– How many expected out of 1000 trials?
– Compare theoretical, empirical results
The Law of Large Numbers
• Probability for each possibility
Roll
Possible ways
Probability
2 or 12
1
1/36 = 0.028
3 or 11
2
2/36 = 0.056
4 or 10
3
3/36 = 0.083
5 or 9
4
4/36 = 0.111
6 or 8
5
5/36 = 0.139
7
6
6/36 = .167
Rolling Two Dice … Final Results
Flipping a Coin … Getting Heads
Flipping a Coin … Final Results
Complement of an Event
• The set of all outcomes in a sample space that
do not include the given event
• Complement of getting heads in a coin flip?
• Complement of rolling a 4 on a single die?
• Complement of rolling a 4 with two dice?
• Complement of getting a hand of 3 cards,
4, 5, 3 ?
• Probability of a complement = 1 – P(E)
Conditional Probability
• Probability of an event, given that another
event has already occurred
• Given drawing a king out of a deck
of cards (and not replaced).
• Now what is the probability of
drawing a queen?


4
P BA 
51
Conditional Probability
• Note that for our question, the “given”
affected the possible outcome of the event
– What is P(draw a queen | king already drawn)
• These are dependent events
• What if we said
– What is P(draw a queen | rolled a 5 on a die)
• These are independent events
P(B | A) = P(B) or P(A | B) = P(A)
Multiplication Rule for P(A and B)
• Probability for two events to occur in
sequence
P( A and B )  P ( A)  P ( B | A)
• P(draw King and draw Queen) =
P(draw King) ∙ P(draw Queen | draw King) =
4 4
  0.006
52 51
Mutually Exclusive Events
• Two events A and B are mutually exclusive
when A and B cannot occur at the same time
Mutually Exclusive Events
• Which of the following are mutually exclusive,
which are not
 A = Randomly select blue sock
B = Randomly select blue piece of clothing
 A = Randomly select a Ford
B = Randomly select a Toyota
Probability P(A or B)YES !
• Recall P(A and B) = P(A) ∙ P(B)
This is when A and B are not mutually
exclusive
• Probability one event or the other
P( A or B) = P(A) + P(B) – P(A and B)
• And if A and B are mutually exclusive, simplify
to
P( A or B) = P(A) + P(B)
Probability P(A or B)
• Example:
P(roll a 6 or roll an odd number)
• Think first … mutually exclusive?
– Yes
• So P ( 6 or odd)
= P(roll a 6) + P(roll odd number)
= 1/6 + ½ = 2/3 = .6667
Link to Apps Shown
Go to:
http://tinyurl.com/hgrbmz6
Download These Slides
• www.letu.edu/people/stevearmstrong
Probability
Webinar Math 201
Liberty University
Dr. Steve Armstrong
[email protected]