Probability, Sample Space and Events

• Probability is a "chance" or likelihood of an event, applicable across mathematics, biology, and business decision-making.
• A statistical experiment is an activity that produces outcomes.
• A sample space is the set of all possible outcomes of a statistical experiment, denoted as S.
• Sample points are elements within a sample space.
• Example: Tossing a coin once S = {H, T}; H and T are sample points.
• Example: rolling a die once S = {1, 2, 3, 4, 5, 6}.

Probability Tree

• A Probability Tree helps determine outcomes of a statistical experiment, branching out to show of all results.
• Example: Tossing a coin twice shows all possible outcomes HH, HT, TH, TT.

Events

• An event is a subset of a sample space.
• Example, if S= {1, 2, 3, 4, 5, 6}, then E= {1} is an event, A= {1, 3, 5} (odd numbers), and B= {2, 4, 6} (even numbers).
• The complement of an event A is all elements of S not in A, denoted A'.
• Example: If S= {1, 2, 3, 4, 5, 6} and A= {1, 3, 5}, then A'= {2, 4, 6}.

Intersection of Events

• The intersection of events A and B contains elements common to both, written as A ∩ B.
• Two events A & B are mutually exclusive (disjoint) if A ∩ B = Ø, with no common elements.
• Two events are mutually exclusive if they cannot occur at the same time

Union of Events

• The union of events A and B, denoted A ∪ B, includes all elements in A, B, or both.
• Example: Outcomes when getting a head or a tail= {H,T}
• Example: Outcomes when getting greater than 3 or less than 4 in a single roll of a die= {1,2,3,4,5,6}

Fundamental Counting Rule

• The fundamental counting rule applies when events are happening concurrently.
• If one event has r₁ outcomes, a second has r₂ outcomes, and so on up to event n with r outcomes, the total outcomes is r₁r₂r₃...*rₙ.
• Example: if you can select from 5 drinks and 4 sandwiches= 5*4 = 20 different possibilities.
• Example: if you can select from four digits 1,2,5,6, and 8= 5⁴ different possibilities= 625.

Permutation

• A permutation is an arrangement of all or part of a set of objects.
• Ex: a,b,c taken two letters leads to ab, ac, bc, ba, ca, cb -> 6 permutations.
• Ex: a,b,c taken two letters leads to ab, ac, bc, -> 3 permutations.

Combination

• A combination is the number of ways to select r objects from n without considering order.
• Factorial notation: n! = n*(n-1)(n-2)...32*1, and o!=1

Permutations Formula

• The permutations of n objects taken r at a time can be found using : n!/(n-r)!
• Example: If there are 26 raffle tickets and 3 are to be drawn for first, second and third prizes. The possible number of sample space S can be found using 26!/(26-3)! = 15,600 ways.

Combinations Formula

• Combinations of n objects taken r at a time: n! / (r!(n-r)!).
• Using the formula, Pₙₙ = n!
• The number of permutations of n distinct objects in a circle = (n −1)!

Cells

• Cells are subsets after dividing a set of n objects where no intersection occurs among pairs of subsets and the union of all subsets equals to the set itself; the number of ways to achieve is: n!/(n₁!n₂!...nₖ!), where n₁ + n₂ + ... + nₖ = n

Probability

• An event's probability, P(A), is the sum of probabilities of simple events constituting A.
• 0 ≤ P(A) ≤ 1, P(0) = 0, and P(S) = 1.
• The probability is derived outcomes that constitute A/total possible outcome.

Theorem 6.4

• If A and B are any two events, then P(A∪B) = P(A) + P(B) - P(A∩B)

Corollary 1

• If A and B are mutually exclusive events, P(A∪B) = P(A) + P(B).

Theorem 6.5

• For 3 events A, B, and C, the probability of their union is P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C).

Corollary 2

• A₁, A₂, A₃,..., Aₙ are Mutually Exclusive or independent events, then P(A₁ ∪ A₂ ∪ A₃ ∪...∪ Aₙ) = P(A₁) + P(A₂) + ... + P(Aₙ)

Theorem 6.6

• If A and A' are complementary events then P(A)+P(A')=1.

Conditional Probability

• It is the probability of an event given that another event has occurred.
• The conditional probability of B, given A [P(B/A)], is P(A∩B)/P(A)
• Two events are independent if the occurrence of one doesn't affect the probability of the other. Statistically, P(B/A) = P(B), P(A | B) = P (A), and P (A ∩ B) = P (A) P (B) if they are independent.

Multiplicative Rule

• the multiplicative rule, if in an experiment the events A and B can both occur then: P(A∩B)=P(A)*P(B/A)

Bayes Rule

• If the events B₁, B₂, B₃,..., Bₖ constitute a partition of the sample space S, such that P(B) ≠ 0 for I = 1, 2, 3, ..., k, then for any event A in S such that P(A) ≠ 0: P (Bᵢ/A) = (P(Bᵢ) P(A/Bᵢ))/(Σᵢ₌₁ᵏ P(Bᵢ) P(A/Bᵢ))