All Formulas

Question 1

How to find P(A) - 4

Answer 1

• Define sample space Ω
• Determine probabilities for each ω in Ω, if all outcomes are equally likely then P(ω) = 1/N
• Determine which outcomes belong to A
• Compute P(A) by Summing all Probabilities of outcomes which belong to A

Question 2

Addition rule - 1

Answer 2

• P(A u B) = P(A) + P(B) - P(A n B)

Question 3

Addition rule disjoint events - 3

Answer 3

• Two events are disjoint if they cannot happen at the same time.
• A n B = ø
• P(A u B) = P(A) + P(B)

Question 4

Complement (and complement rule) - 2

Answer 4

• The complement of A (A bar or Ac) is the set of outcomes which is not in A
• P(A bar) = 1 - P(A)

Question 5

Conditional probability - 3

Answer 5

• The probability of B occurs after A has occurred
• P(B|A) = P(A n B) / P(A)
• !!! P(B|A) ≠ P(A|B)

Question 6

Multiplication rule - 1

Answer 6

• P(A n B) = P(A) * P(B|A)

Question 7

Independence - 2

Answer 7

• Two events A and B are independent if P(A n B) = P(A) * P(B)
• Independence ≠ Disjointness

Question 8

Complement of at least one - 1

Answer 8

• P(at least one occurence of … ) = 1 - P(no occurence of … )

Question 9

Simple law of total probability - 2

Answer 9

• A and B are two events

- P(B) = P ( B ∩ A ) + P ( B ∩ A bar ) = P ( B | A ) · P ( A ) + P ( B | A bar ) · P ( A bar )

Question 10

Bayes’ Theorem - 3

Answer 10

• P(A|B)= P(B|A)·P(A) / P(B|A)·P(A)+P(B|A bar)·P(A bar).
• P(B|A) + P(B bar | A) = 1
• P(B|A) + P(B|A bar ) ≠ 1

Question 11

Law of total probability - 2

Answer 11

• A1, … , Am is a partition

- P(B) = Sum(m, i = 1) P(B ∩ Ai ) = Sum(m, i = 1) P(B|Ai ) · P(Ai ).

Question 12

Find probability distribution of discrete random variable - 5

Answer 12

• Determine sample space of probability experiment and probabilities of each outcome
• List numerical values X(ω) for ω ∈ Ω
• For each numerical value x of X find collection of simple events which have particular numerical value x {X = x} = {ω : X(ω) = x}
• use P({w}) determine probability of event {X = x}
  P(X =x)= P({ω : X(ω) =x})= sum(ω:X (ω)=x) P({ω}
• Tabulate results with left column with all values x of X and a column with probabilities P(X = x)

Question 13

Expected value (expectation/mean) - 3

Answer 13

• X is a discrete random variable
• Expected value of X is weighted average of the possible values of X
• μ = sum(k, i = 1) xi * P(X = xi)

Question 14

Variance and SD of X - 2

Answer 14

• σ^2 =Var(X) = sum(k, i = 1) (xi −μ)^2 2 * P (X =xi).

- σ = SD(X) = √Var(X)

Question 15

Normal distribution formula - 1

Answer 15

• p(x) = 1 / σ(√2π) * e^(-(1/2) * ( (x-μ) /σ )^2

Question 16

Relation between general normal distribution and standard normal - 2

Answer 16

• If X is normally distributed (mean μ and standard deviation σ) then a shifted and rescaled version Z has a standard normal distribution
• Z = (X - μ) / σ

Question 17

z score value of x - 2

Answer 17

• Number of standard deviations away from the mean

- z = (x - μ) / σ

Question 18

Central limit theorem - 3 + 2

Answer 18

• Sample of size n > 30 from population with mean μ and standard deviation σ
• For normal population the sample can be of any size, instead of n > 30
• X Bar, or sample mean has approximately a normal distribution
• • mean = μ
• • Standard deviation = σ / √n