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First lecture
Faten alamri
• “probability is the most important
concept in modern science especially
as nobody has slightest notation what it
means”
• Counting techniques
• 1) multiplication rule
3)permutation
2) computation
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• Old saying
P(A) = N(A)\N(S)
Note: these are called “counting methods” because
we have to count the number of ways A can occur
and the number of total possible outcomes.
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# of ways A can occur
P( A) 
total # of outcomes
# of aces in the deck
4
P(draw an ace) 

 .0769
# of cards in the deck 52
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Example 1: You draw one card from a deck of
cards. What’s the probability that you draw an
ace?
Toss 1:
2 outcomes
H
Toss 2:
2 outcomes
H
T
T
22 total possible outcomes: {HH, HT,
TH, TT}
H
T
1 way to get HH
P( HH )  2
2 possible outcomes
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• Example
• What's the probability of 2 heads when tossing a coin?
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Example
• What’s the probability of rolling a die then tossing a
coin?
Permutations—
order matters!
Combinations—
Order doesn’t
matter
With replacement
Without replacement
Without replacement
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Counting methods for computing probabilities
• Example
• How many license plates can we have using three
letters followed by three digits ?
• Alphabet 26 digit # 10
• = ( 26)3 (10)3
• =(17576)(1000)
• =17,576,000
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• Formally, “order matters” and “with replacement”
use powers
• Example 1.6 pg5
• Example
• From a club of 24 members, a President, Vice
President, Secretary, Treasurer and Historian are to
be elected. In how many ways can the offices be
filled?
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• Permutation: The number of ways in which a subset of
objects can be selected from a given set of objects,
where order is important
24!
24!


24 p5 
( 24  5)! 19!
24 * 23 * 22 * 21 * 20  5,100,480
3
Example 1.4. How many permutations are there of all three of letters a, b,
and c?
Answer:
p3= n!\(n − r)!
=3!\0!
=6
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• Answer
Combinitions
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• Combination: The number of ways in which a
subset of objects can be selected from a given set
of objects, where order is not important.
• A Combination is an arrangement of items in
which order does not matter.
Since the order does not matter in combinations, there are
fewer combinations than permutations. The combinations
are a "subset" of the permutations.
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To find the number of Combinations of n
items chosen r at a time, you can use the
formula
c is denoted by
n
c   
r 
n
n!
  
 r  (n  r )r!
52!
52!


52 C5 
5!(52  5)! 5!47!
52 * 51* 50 * 49 * 48
 2,598,960
5 * 4 * 3 * 2 *1
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• Example 1.7 pg 5
• Example
• To play a particular card game, each player is dealt five
cards from a standard deck of 52 cards. How many
different hands are possible?
• Answer
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Guidelines on Which Method to Use