Test 1 Formulae/Definitions/Rules

Question 1

Theorem - Counting rule for compound events

Answer 1

N1 x N2 x … x Nk

Question 2

Theorem - Permutations of n distinct objects

Answer 2

n!

Question 3

Theorem - Permutations of n objects taken r at a time

Answer 3

nPr = n!/(n-r)! - order matters

Question 4

Theorem - Circular Permutations

Answer 4

n!/n=(n-1)!

Question 5

Theorem - Permutations of repeated elements

Answer 5

n!/ n1! x n2! x…x nk!

Question 6

Theorem - Combinations of the binomial theorem

Answer 6

( n choose r) = nPr/r! = n!/r!(n-r)! - order does not matter

Question 7

Theorem - Binomial theorem

Answer 7

(x+y)^n = (n choose r) x^n-r y^r
or
(n choose n-r)

Question 8

Theorem - Pascal’s triangle coefficients

Answer 8

(n choose r) = (n-1 choose r) + (n-1 choose r-1)

Question 9

Theorem - multinomial coefficients

Answer 9

( n choose r1, r2, r3, … , rk) = n!/r1! x r2! x … rk!

Question 10

Theorem - Probability of an event with equal outcomes

Answer 10

P(A) = n (possible or optimal outcome)/k (all outcomes)

Question 11

DeMorgan’s Law

Answer 11

(A’uB’) = A’ n B’

Question 12

Postulates of Probability

Answer 12

P1. Non-negative real number
P2. P(S)=1
P3. P(A1uA2…)=P(A1)+P(A2)…
- countably additive
- if all three are satisfied a probability can be assigned

Question 13

Theorem - Probability of an event

Answer 13

If A is an event in S, P(A) is the sum of the individual outcomes of A

Question 14

Rules of Probability

Answer 14

1. P(A) + P(A’) = 1 or P(A’)=1-P(A)
2. P(null)=0
3. If AcB than P(A) less than or equal to P(B)
4. 0 less than or equal to P(A) less than or equal to 1
5. Inclusion-Exclusion Principles

Question 15

Theorem - Inclusion-Exclusion Principle

Answer 15

P(AuB) = P(A) + P(B) - P(AnB)
- for three sets:
P(AuBuC)=P(A) + P(B) +P(C) - P(AnB) - P(AnC) - P(BnC) + P(AnBnC)

Question 16

Theorem - Conditional Probability

Answer 16

P(A|B) = P(AnB)/P(B)

Question 17

Theorem - Conditional Probability Multiplication Rule

Answer 17

P(A) x P(B|A) = P(BnA)
P(AnBnC)=P(A) x P(B|A) x P(C|AnB)

Question 18

Definition - Independent and Dependent events

Answer 18

Events A and B are independent only if P(AnB)=P(A) x P(B) otherwise they are dependent
- If A and B are independent, then so are A and B’
- Additionally, A1, A2 etc are independent if the probability of the intersection of any number of these is equal to the product of their individual probabilities

Question 19

Rule of total probability

Answer 19

P(V)=P(A)·P(V|A)+P(B)·P(V|B)