Mathematics

How to Calculate and Solve for General Binomial Distribution | Probability

January 7, 2020 by Loveth Idoko

The image above represents general binomial distribution.

To compute for general binomial distribution, four essential parameters are needed and these parameters are n, r, q and q.

The formula for calculating general binomial distribution:

P(r successes) = ^n!⁄_{(n – r)!r!}q^{(n – r)}p^r

Where;

P(r successes) = General Binomial Distribution
p = P(A)
q = P(not A)

Let’s solve an example;
Find the general binomial distribution when n is 8, r is 6, p is 1 and q is 0.

This implies that;

n = 8
r = 6
p = 1
q = 0

P(r successes) = ^n! ⁄ _{(n – r)!r!}q^{(n – r)}p^r
P(6 successes) = ^8! ⁄ _{(8 – 6)!6!}(0)^{(8 – 6)}(1)⁶
P(6 successes) = ^8! ⁄ _2!6!(0)²(1)⁶
P(6 successes) = ⁴⁰³²⁰ ⁄ _(2)(720)(0)²(1)⁶
P(6 successes) = ⁴⁰³²⁰ ⁄ ₁₄₄₀(0)²(1)⁶
P(6 successes) = (28)(0)²(1)⁶
P(6 successes) = (28)(0)(1)
P(6 successes) = 0

Therefore, the general binomial distribution is 0.

How to Calculate and Solve for Dependent Events | Probability

January 6, 2020 by Loveth Idoko

The image above represents dependent events.

To compute for dependent events, four essential parameters are needed and these parameters are Number of Times Event A can occur (x_A), Number of Times Event B can occur (x_B) and Total Number of All Possible Outcomes (N).

The formula for calculating independent events:

P(A and B) = P(A) x P(B|A)

Where;

P(A and B) = Dependent events
^x_A = Number of Times Event A can occur
^x_B = Number of Times Event B can occur
N = Total Number of All Possible Outcomes
P(A) = ^x_A ⁄ _N
P(B|A) = ^x_B ⁄ _{(N – 1)}

Let’s solve an example;
Find the dependent events when the number of times event A can occur is 8, number of times event B can occur is 11 and the total number of all possible outcomes is 18.

This implies that;

^x_A = Number of Times Event A can occur = 8
^x_B = Number of Times Event B can occur = 11
N = Total Number of All Possible Outcomes = 18

P(A and B) = P(A) x P(B|A)
P(A and B) = ^x_A ⁄ _N x ^x_B ⁄ _{(N – 1)}
P(A and B) = ⁸ ⁄ ₁₈ x ¹¹ ⁄ ₁₇
P(A and B) = ^(8)(11) ⁄ _(18)(17)
P(A and B) = ⁸⁸ ⁄ ₃₀₆
Dividing the numerator and denominator by 2
P(A and B) = ⁴⁴ ⁄ ₁₅₃
P(A and B) = 0.287

Therefore, the dependent events is 0.287.

How to Calculate and Solve for Independent Events | Probability

January 6, 2020 by Loveth Idoko

The image above represents independent events.

To compute for independent events, four essential parameters are needed and these parameters are Number of Times Event A can occur (x_A), Number of Times Event B can occur (x_B) and Total Number of All Possible Outcomes (N).

The formula for calculating independent events:

P(A and B) = P(A) x P(B)

Where;

P(A and B) = Independent events
^x_A = Number of Times Event A can occur
^x_B = Number of Times Event B can occur
N = Total Number of All Possible Outcomes
P(A) = ^x_A ⁄ _N
P(B) = ^x_B ⁄ _N

Let’s solve an example;
Find the independent events when the number of times event A can occur is 11, number of times event B can occur is 15 and the total number of all possible outcomes is 22.

This implies that;

^x_A = Number of Times Event A can occur = 11
^x_B = Number of Times Event B can occur = 15
N = Total Number of All Possible Outcomes = 22

P(A and B) = P(A) x P(B)
P(A and B) = ^x_A ⁄ _N x ^x_B ⁄ _NP(A and B) = ¹¹ ⁄ ₂₂ x ¹⁵ ⁄ ₂₂P(A and B) = ^(11)(15) ⁄ _(22)(22)P(A and B) = ¹⁶⁵ ⁄ ₄₈₄Dividing the numerator and denominator by 11
P(A and B) = ¹⁵ ⁄ ₄₄P(A and B) = 0.3409

Therefore, the independent events is 0.3409.

How to Calculate and Solve for Mutually Non-Exclusive | Probability

January 6, 2020 by Loveth Idoko

The image above represents mutually non-exclusive.

To compute for mutually non-exclusive, four essential parameters are needed and these parameters are x_A, N_A, x_B and N_B.

The formula for calculating mutually non-exclusive:

P(A or B) = P(A) + P(B) – P(A and B)

Where;

P(A or B) = Mutually Non-Exclusive
P(A) = ^x_A ⁄ _N_A
P(B) = ^x_B ⁄ _N_B

Let’s solve an example;
Find the mutually non-exclusive when the x_A is 10, N_A is 20, x_B is 5 and N_B is 12.

This implies that;

x_A = 10
N_A = 20
x_B = 5
N_B = 12

P(A or B) = P(A) + P(B) – P(A and B)
P(A or B) = P(A) + P(B) – (P(A) x P(B))
P(A or B) = ^x_A ⁄ _N_A + ^x_B ⁄ _N_B – (^x_A ⁄ _N_A x ^x_B ⁄ _N_B)
P(A or B) = ¹⁰⁄ ₂₀ + ⁵⁄ ₁₂ – (¹⁰⁄ ₂₀ x ⁵⁄ ₁₂)
P(A or B) = ^{10(12) + 5(20)}⁄ _(20)(12) – (^(10)(5)⁄ _(20)(12))
P(A or B) = ^{120 + 100}⁄ ₂₄₀ – (⁵⁰⁄ ₂₄₀)
P(A or B) = ²²⁰⁄ ₂₄₀ – ⁵⁰⁄ ₂₄₀
P(A or B) = ^{(220 – 50)}⁄ ₂₄₀
P(A or B) = ¹⁷⁰⁄ ₂₄₀
Dividing the numerator and denominator by 10
P(A or B) = ¹⁷⁄ ₂₄
P(A or B) = 0.708

Therefore, the mutually non-exclusive is 0.708.