Probability

Probability

Sample Space and Relationships among Events

• Sample Space (S):

  • A set of all possible outcomes of a random process.

  • Denoted as S.

  • Example: for a die toss, S = {1, 2, 3, 4, 5, 6}.

    • If considering odd or even numbers, S = {even, odd}.

Events

• Event:

  • Any subset of a sample space.

  • Collection of elements from a sample space.

  • Denoted using capital letters (e.g., A, B, C).

  • Example: For a quality check on manufactured items, S = {defective, nondefective}.

    • Event A = Observing a defective item.

More Complex Events

• In inspecting a sample of 25 items, the sample space for defective items classified asS = {0, 1, 2, 3, ..., 25}.

  • Events can be defined as:

    • A = Observing at most 2 defective items (A = {0, 1, 2}).

    • B = Observing at least 2 but no more than 7 defective items (B = {2, 3, 4, 5, 6, 7}).

Equally Likely Events

• Events are considered equally likely if one does not occur more often than another.

• Tree diagrams can represent sample spaces.

  • Example: Prizes at a game are awarded by randomly chosen ticket stubs.

    • Each attendee has an equal chance of winning.

  • In a raffle scenario, purchasing more tickets increases a person’s chances of winning.

Tree Diagram Example

• For a decision to build two plants in either eastern cities (A, B, C, D) or western cities (E, F):

  • The sample space formed by pairing eastern and western cities is:S = {AE, AF, BE, BF, CE, CF, DE, DF}.

Venn Diagrams

• Venn diagrams help display sample spaces and relationships among events.

  • Example: Throwing a 6-sided die with events defined as:

    • A = An even number (A = {2, 4, 6}).

    • B = An odd number (B = {1, 3, 5}).

    • C = A number greater than 4 (C = {5, 6}).

    • E3 = Observing face with number 3 (E3 = {3}).

Calculating Probabilities

• Probability of Events:

  • If events are equally likely, probability is the relative frequency (P(event) = h/n).

• Example based on 100 electronics engineering majors who take calculus or signal processing.

  • Example Events:

    • A = Students in calculus (30 students).

    • B = Students in signal processing (25 students).

    • Probability of both courses, neither course, etc., are calculated based on these counts.

Complement and Combined Events

• Complement of Events:

  • The complement of event A is all outcomes not in A. Denoted as A'.

  • Union (A ∪ B): outcomes in at least one of A or B.

  • Intersection (A ∩ B): outcomes in both A and B.

Mutually Exclusive Events

• Two events are mutually exclusive if they cannot occur at the same time.

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

Counting Rules

• Product Rule: If task one has n1 outcomes and task two has n2 outcomes, total outcomes = n1 * n2.

• Permutations: Arrangements of n objects taken r at a time (e.g. A = arrangements of employees).

• Combinations: Ways to choose r objects from n, order does not matter.

Probability Rules

• Complementary Events: Probability of not A is 1 - P(A).

• Additive Rule: For mutually exclusive events, P(A ∪ B) = P(A) + P(B).

• Multiplicative Rule: For independent events, P(A ∩ B) = P(A) * P(B).

Example Scenarios

• Calculate probabilities for various combinations of defective items, enrollment overlap in courses, etc.

• Use provided data (number of students or defective items) to illustrate probability examples.