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Probability and combinatorics
Notes from www.math.uah.edu/stat/
Virtual Laboratories in Probability and Statistics
Discrete probability distributions
List of mutually exclusive and exhaustive outcomes of
some process and their probabilities
Example - 1 coin toss
Outcome:
Head Tail
Probability: 1/2
1/2
Example - 1 fair die throw
#spots:
1
2
3
4
5
6
Probability: 1/6 1/6 1/6 1/6 1/6 1/6
This is an example of a discrete uniform distribution
Uniform distribution and counting
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Atoms
1
2
.
.
N
Probability: 1/N 1/N 1/N 1/N 1/N 1/N
•P(A) = N(A)/N
•e.g. A = get less than 3 when you roll a dice
•P(A) = P(1) + P(2) = 2/6
•So calculation of probabilities  counting
•But counting can be complicated. Need rules
Multiplication rule for ordered events
• Flying from LA to NY and back
• How many airline combinations are possible?
• E.g. (American, United), (United, American),
(United, United)
• These are called ordered pairs
• Rule: If the first element of the pair can
happen in n1 ways and for each of these
ways, the second element can happen n2
ways, then the total no. of possibilities is
n 1 n2
Probabilities using product rule
• A license number consists of two letters (uppercase)
followed by five digits.
1. How many different license numbers are there?
2. If a license number is chosen at random, find the
probability that the digits are all less than 5.
• Total license nos. = 26*26*105
• Valid license nos.=26*26*45
• P(all digits < 5) = Valid license nos./Total license nos.
•
= (4/10)5 = 0.01024
Obstetricians and paediatricians
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Family needs to choose from 2 clinics:
Clinic 1 has 2 obs and 2 paed
Clinic 1 has 2 obs and 3 paed
How many combos are possible?
Unequal nos. of possibilities
Tree diagrams
P1
P2
P1
O1
O2
P1
O3
Total ways = 2+2+3+3
P3
O4
P4
P5
P3
P4
P5
Permutations
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Sample space S with n elements
Want to choose an ordered sequence of k
No of ways possible= nPk
nPk= n(n-1)(n-2)…(n-k+1)
In statistics this is called sampling without
replacement
• A special form of pigeonhole principle
Arrangement problem
Twelve books, consisting of 5 math books, 4 science books
and 3 history books are randomly arranged on a bookshelf.
1. How many outcomes are there?
2. Find the probability that the books of each type are together.
3. Find the probability that the math books are together.
Combinations
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Consider a sample space S with n
elements. A combination of size k from S
is an (unordered) subset {x1, x2, ..., xk} of
S with k (distinct) elements
C(n,k) = nPk/k!
This is also sampling without replacement,
but order is ignored
Games of chance
A poker hand consists of 5 cards dealt without
replacement and without regard to order from a
deck of 52 cards.
1. Show that the number of poker hands is
2,598,960.
2. Find the probability that a random poker hand is
a full house (3 cards of one kind and 2 of
another kind) 0.001440576
3. Find the probability that a random poker hand
has 4 of a kind. 0.000240096
Independence
• Events X and Y are said to be independent if
the probability that X occurs is not affected by
whether or not Y has occurred.
• Examples of independent events:
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It can be shown that independence implies:
P(X and Y) = P(X)P(Y)
(and not otherwise)
This is called the multiplication rule for
independent events.
Example: 2 security guards
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Guard 1: answers pager 80% of the time
Guard 2: answers pager 50% of the time
What is the probability
i) they both respond to a call?
ii) at least one responds to a call?
X= event guard 1 answers; Y= guard 2
answers
• What can we say about X and Y?
Tree models to calculate probabilities
DISCRETE DISTRIBUTIONS
Example - Family of 3 children.
Let X = number of girls
Possible values:
X = 3
GGG
X = 2
GGB
GBG
BGG
X = 1
BBG
BGB
GBB
X = 0
BBB
Assume the 8 outcomes are equally likely so that
x
0
1
2
3
P(X = x)
1/8
3/8
3/8
1/8
P(X  x)
Probability Distribution of a Discrete r.v.
• The probabilities may be written as:
P( X  xi )  pi for i  1,2,..., k and 0  pi  1
k
 p 1
i
i 1
•P(Xi=xi) is also referred to as the density function f(x)
•The cumulative distribution function (c.d.f.) is defined as
i
F ( xi )  P( X  xi )   p j
j 1
Example - Bernoulli trials
Each trial is an 'experiment' with exactly 2 possible outcomes,
"success" and "failure" with probabilities p and 1-p.
Let X = 1 if success, 0 if failure
Probability distribution is
x
P(X = x)
0
p
1
1-p
• Results for Bernoulli trials can be simulated using S-PLUS
• e.g. simulate results of a drug trial drug, success (cure) has
probability p = 0.3 for each patient, 100 patients in trial.
• result _ rbinom(100, size=1, prob=p)
• result is a 100 vector that looks like 1,0,0,1,0,1,…...
Example-Binomial Experiment
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Generalisation of Bernoulli trials
X ~ Bin(n,p)  X = # of successes in n Bernoulli trials
e.g. X = # of heads in 10 tosses of a coin, n = , p =
e.g. X = # of boys in a family of 5 children, n= , p =
e.g. X = # of sixes in 100 rolls of a dice, n= , p=
possible values for X =
probability distribution for X
(q = 1-p)
n
P(X = k) =   p k q n  k
k 
 
 n  k nk
  p q 

k 0  k 
n
Shape of the Binomial Distribution
• The shape of the binomial distribution depends on
the values of n and p.
• probdistr_ dbinom(x=0:n, size=n, prob=p)
Prob. Distr. for Bin(10,0.5)
0.0
0.0
0.05
0.05
0.10
0.10
0.15
0.15
0.20
0.20
0.25
0.25
Prob. Distr. for Bin(15,0.8)
0
2
4
6
no. of successes
8
10
0
5
10
no. of successes
15
Example - 2 dice are thrown
Let X denote the sum of the results. Outcomes:
11
21
31
41
51
61
12
22
32
42
52
62
13
23
33
43
53
63
14
24
34
44
54
64
15
25
35
45
55
65
16
26
36
46
56
66
Assume the 36 outcomes are equally likely so each has probability = 1/36
Possible values of X are 2, 3, ... , 12
e.g. P(X = 4) = P(1,3 or 2,2 or 3,1) = 3/36 .
The probability distribution is
x
2
P(X=x) 1/36
3
4
2/36
3/36
...
...
10
11
12
3/36
2/36
1/36