Probability
1. Permutations and Combinations
Permutations : nPm
Combinations : nCm
Whole numbers : n , m
1. Factorial
n = 1 2 3 … (n – 2)(n – 1)n
0 =1
2.
n
3.
n
Pn = n
Pm =
4. Binomial Coefficient
n
Cm =
5.
n
6.
n
7.
n
=
Cm = nCn-m
Cm + nCm+1 = n+1Cm+1
C0 + nC1 + nC2 + … + nCn = 2n
8. Pascal’s Triangle
Row 0
Row 1
Row 2
Row 3
Row 4
Row 5
Row 6
1
1
1
1
1
1
1
2
3
4
5
6
1
3
6
10
15
1
1
4
10
20
1
5
15
1
6
1
2.
Probability Formulas
Events : A, B
Probability : P
Random variables : X , Y , Z
Values of random variables : x , y , z
Expected value of X :
Any positive real number :
Standard deviation :
Variance :
Density functions : f(x) , f(t)
9.
Probability of an Event
P(A) =
,
where
m is the number of possible positive outcomes,
n is the total number of possible outcomes.
10.
0
P(A)
11.
Range of Probability Values
1
Certain Event
P(A) = 1
12.
Impossible Event
P(A) = 0
13.
Complement
P( ) = 1 – P(A)
14.
Independent Events
P(A / B) = P(A) ,
P(B / A) = P(B)
15.
P(A
Addition Rule for Independent Events
B ) = P(A) + P(B)
P(A
Multiplication Rule for Independent Events
B) = P(A) P(B)
P(A
General Addition Rule
B) = P(A) + P(B) – P(A B),
16.
17.
where
A B is the union of events A and B,
A
B is the intersection of events A and B.
18.
Conditional Probability
P(A / B) =
19.
P(A
20.
Law of Total Probability
,
P(A) =
B) = P(B) P(A / B) = P(A) P(B / A)
where Bi is a sequence of mutually exclusive events.
21. Bayes’ Theorem
P(B / A) =
22. Bayes’ Formula
P(Bi / A) =
where
Bi is a set of mutually exclusive events (hypotheses),
A is the final event,
P(Bi) are the prior probabilities,
P(Bi / A) are the posterior probabilities.
23.
Law of Large Numbers
P(
)
0 as n
,
P(
)
1 as n
,
where
Sn is the sum of random variables,
n is the number of possible outcomes.
24. Chebyshev Inequality
P(
)
,
where V(X) is the variance of X.
25. Normal Density Function
(x) =
,
where x is a particular outcome.
26. Standard Normal Density Function
(z) =
Average value
= 0 , deviation
= 1.
Figure 1.
27. Standard Z value
Z=
28. Cumulative Normal Distribution Function
dt ,
F(x) =
where
x is a particular outcome,
t is a variable of integration.
29. P(
) = F(
) - F(
),
where
X is normally distributed random variable,\
F is cumulative normal distribution function,
P(
) is interval probability.
30. P(
) = 2F( ),
where
X is normally distributed random variable,
F is cumulative normal distribution function.
31. Cumulative Distribution Function
,
F(x) = P(X x) =
where t is a variable of integration.
32. Bernoulli Trials Process
= np ,
= npq ,
where
n is a sequence of experiments,
p is the probability of success of each experiments
q is the probability of failure, q = 1 – p.
33. Binomial Distribution Function
pkqn-k ,
B(n , p , q) =
= np ,
= npq ,
f(x) = (q + pex)n ,
where
n is the number of trials of selections,
p is the probability of success,
q is the probability of failure, q = 1 – p.
34. Geometric Distribution
P(T = j) = qj-1p,
=
,
,
=
where
T is the first successful event is the series,
j is the event number,
p is the probability that any one event is successful,
q is the probability of failure, q = 1 – p.
35. Poisson Distribution
,
P(X = k)
,
=
= np,
,
where
is the rate of occurrence,
k is the number of positive outcomes.
36. Density Function
P(a
X
b) =
37. Continuous Uniform Density
f=
,
,
=
where f is the density function.
38. Exponential Density Function
,
f(t) =
,
=
is the failure rate.
where t is time,
39.
Exponential Distribution Function
F(t) = 1 -
,
where t is time,
40.
is the failure rate.
Expected Value of Discrete Random Variables
= E(X) =
,
where xi is a particular outcome, pi is its probability.
41.
42.
Expected Value of Continuous Random Variables
= E(X) =
Properties of Expectations
E(X + Y) = E(X) + E(Y) ,
E(X - Y) = E(X) - E(Y) ,
E(cX) = cE(X),
E(XY) = E(X) E(Y),
where c is a constant.
43.
E(X2) = V(X) +
where
= E(X) is the expected value,
= is the variance.
44.
Markov Inequality
P(X > k)
,
where k is some constant.
45.
Variance of Discrete Random Variables
= V(X) = E[(X – )2] =
,
where
xi is a particular outcome,
pi is its probability.
46.
47.
Variance of Continuous Random Variables
= V(X) = E[(X - )2] =
Properties of Variance
V(X + Y) = V(X) + V(Y),
V(X - Y) = V(X) + V(Y),
V(X + c) = V(X)
V(cX) = c2V(X),
where c is a constant.
48.
Standard Deviation
=
D(X) =
49.
Covariance
cov(X, Y) = E[(X – (X))(Y – (Y))] = E(XY) - (X) (Y) ,
where
X is random variable,
V(X) is the variance of X,
is the expected value of X or Y.
50.
Correlation
(X ,Y) =
,
where
V(X) is the variance of X,
V(Y) is the variance of Y.