# All Probability-and-estimation Formulas List

## Normal Probability

#### Single Event Probability Formula :

Probability of event A that occurs P(A) = n(A) / n(S).
Probability of event A that does not occur P(A') = 1 - P(A).

#### Multiple Event Probability Formula :

P(A) = n(A) / n(S).
P(A') = 1 - P(A).
P(B) = n(B) / n(S).
P(B') = 1 - P(B).
Probability that both the events occur P(A ∩ B) = P(A) x P(B).
Probability that either of event occurs P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Conditional Probability P(A | B) = P(A ∩ B) / P(B).

Where,

n(A) - Number of Occurrence in Event A,
n(B) - Number of Occurrence in Event B,
n(S) - Total Number of Possible Outcomes.

## Bayesian Statistics and Analysis

#### Formulas:

Sensitivity = TP / ( TP + FN )
Specificity = TN / ( FP + TN )
Predictive Value Positive = TP / ( TP + FP )
Predictive Value Negative = TN / ( FN + TN )
Positive Likelihood = Sensitivity / ( 1 - Specificity )
Negative Likelihood = ( 1 - Sensitivity ) / Specificity

Where,

TP = True Positive.
FP = False Positive.
FN = False Negative.
TN = True Negative

## Confidence Interval Variance

#### Formula

Standard Deviation Confidence Interval for Variance = [(n-1)×S² / χ²α/2, n-1] ≤ σ² ≤ [(n-1)×S² / χ²1-α/2, n-1]

Where,

n = Sample Size
S = Variance
α = 1 - (Confidence Level/100)
χ²α/2, n-1 = χ²-table value

## Trend Line

#### Formula :

Trend Line Equation(y) = a + bx

Where,

Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
Intercept(a) = (ΣY - b(ΣX)) / N

## Beta Function for Probability Density

Formula Used:
pdf = Where,
pdf - Probability Density Function
Β - Beta Function
x - Probability Distribution Intervals (0≤x≤1)
α and β - Positive shape parameters

## Kurtosis and Skewness Statistics

Formula Used: Where, is the mean,
s is the Standard Deviation,
N is the number of data points.

## Harmonic Mean Frequency

Formula:
Harmonic mean = where,
d is the data values and
f is the corresponding frequency values.

## Circular Permutations

Formula:
Circular Permutation (Pn) = (n-1)!

## Relative Risk Confidence Interval

#### Formula:

Relative Risk (RR) = ( a / ( a + b ) ) / ( c / ( c + d ) )

Where,

a = Exposed Group Positive Outcome,
b = Exposed Group Negative Outcome,
c = Control Group Positive Outcome,
d = Control Group Negative Outcome,

## Combination with Replacement

Formula:
CR(n,r) = C(n+r-1,r) = (n+r-1)! / r! (n - 1)! For n >= 0, and r >= 0.

## Permutation with Replacement

Formula:
PR(n,r) = nrFor, n >= 0, and r >= 0.

## Odd and Even Permutation

#### Formula:

Odd Permutation = n! / 2 ∀ n>=2
Even Permutation = n! / 2 ∀ n>2

## Counting Permutations With Repetition

Formula:
nPr=nr

Where,
n is the number of types,
r is the number (of times) to be chosen.

## Counting Combinations With Repetition

Formula: Where,
n is the number of types,
r is the number (of times) to be chosen.

## Confidence Interval of Reference Range Limit

#### Formula:

Lower Limit = m - t0.975,n-1 x √((n+1)/n) x s.d
Upper Limit = m + t0.975,n-1 x √((n+1)/n) x s.d
SDSRRL = s.d / 2
LlciLlrr = Llrr - t0.975,n-1 x √((n+1)/n) x SDSRRL
UlciLlrr = Llrr + t0.975,n-1 x √((n+1)/n) x SDSRRL
LlciUlrr = Ulrr - t0.975,n-1 x √((n+1)/n) x SDSRRL
UlciUlrr = Ulrr + t0.975,n-1 x √((n+1)/n) x SDSRRL

Where,

t - Distribution.
s.d - Standard Deviation
n - Total Count
m - Mean
LlciLlrr - is the Lower limit of the confidence interval of the Lower limit of the standard reference range
UlciLlrr - is the Upper limit of the confidence interval of the Lower limit of the standard reference range
LlciUlrr - is the Lower limit of the confidence interval of the Upper limit of the standard reference range
UlciUlrr - is the Upper limit of the confidence interval of the Upper limit of the standard reference range
SDSRRL - is the standard deviation of the standard reference range limit
Llrr - is the Lower limit of the standard reference range
Ulrr - is the Upper limit of the standard reference range

## Margin of Error

#### Formula:

M = C x E

Where,

C = Critical Value
E = Standard Error of the Statistics
M = Margin of Error

## Chebyshev Inequality

Formula :
Probability P(X-μ<2σ) = 1 - (1/K2)

Where,

K = Standard Deviation

## Residual Sum of Squares

#### Formula: Where,

X,Y - set of values
α , β - constant values
n - Set value counts

## Probability of Success

Formula:
P (X = r) = nCr pr (1-p)n-r

Where,

Combination nCr = n! / ((n- r)! × r!)
n = number of events
p = Probability of success for each trial
r = 0, 1, ... n

## Pooled Standard Deviation

#### Formula:

Pooled Standard Deviation = √(((n1-1) × s1²+(n2-1) × s2²+...+(nk-1) × sk²) / (n1+n2+...+nk-k))

Mean = Sum of data values / N(Number of Values) Where,

n1,n2,...nk= sample size of each group
s1,s2,...sk= Standard deviation of each group
k=Number of group
M=Mean

## Even Permutation

#### Formula:

Even Permutation = n! / 2, ∀ n>2

## Odd Permutation

#### Formula:

Odd Permutation = n! / 2, ∀ n>=2

## Same Birthday Probability

#### Formula:

p(n) = 1 - (365! / (365n x (365 - n)!))

Where,

n = Number of Persons in a Group
p(n) = Probability with Same Birthdays