Unit 4 - Probability

Probability

• Definition: Probability is a measure of the degree of chance or likelihood that an uncertain event will occur.

  • It reflects the strength of belief that an event will happen.

  • Probability values range from 0 (impossible) to 1 (certain).

  • Can also be expressed as percentages (0% to 100%) or fractions.

Learning Objectives

• Compute basic probabilities.

• Apply the laws of probability.

• Define and solve problems using addition, multiplication, and conditional probability rules.

• Understand independent, mutually exclusive, and mutually exhaustive events.

• Prove the independence of two events.

Interpreting Probabilities

• 0 or 0%: The event cannot occur.

• 1 or 100%: The event must occur.

• Probabilities near 0 indicate a low likelihood of occurrence.

• Probabilities near 1 indicate a high likelihood of occurrence.

Basic Probability Concepts

• Probability formula:

  • ( P(X) = \frac{\text{Total Number of Specified Outcomes}}{\text{Total Number of Possible Outcomes}} )

• Relative Frequency Example: Pick a random student from a class of 28 girls and 12 boys.

  • ( P(girl) = \frac{28}{28 + 12} = \frac{28}{40} = 0.7 ) or 70%.

Complement

• Complement of an event A, noted as ( A^c ), represents all outcomes not equal to A.

• If A represents all nursing students, then ( A^c ) is all students who are NOT nursing students.

Complement Formula

• ( P(A^c) = 1 - P(A) )

  • Example: If ( P(A) = 0.32 ) or 32%, then ( P(A^c) = 1 - 0.32 = 0.68 ) or 68%.

  • Relationship: ( P(A) + P(A^c) = 1 ).

Intersection

• The intersection of events A and B (notated as ( A \cap B )) includes all outcomes that belong to both A and B.

• Look for keywords like "and," "both," or "joint occurrence."

Union

• The union of events A and B (notated as ( A \cup B )) includes all outcomes belonging to either A or B.

• Look for keywords like "either," "or," or "at least."

• Probability formula: ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ).

Mutually Exclusive Events

• Events that:

  • Have no elements in common.

  • Have no intersection.

  • Cannot occur simultaneously.

• Formula: ( P(A \cap B) = 0 ).

Independent Events

• Events A and B are independent if the occurrence of one does not affect the other.

• Probability remains unchanged regardless of the occurrence of the other event.

Independent Events Formula

• For independent events:

  • ( P(A \cap B) = P(A) \times P(B) )

  • ( P(A | B) = P(A) ) (not affected by B)

  • ( P(B | A) = P(B) ) (not affected by A).

Independent Events Example

• If the probability a student takes ECON1005 is 12% and SOCI1005 is 15%,

• Probability both courses are taken:

  • ( P(E \cap S) = P(E) \times P(S) = 0.12 \times 0.15 = 0.018 ) or 1.8%.

• Probability at least one course is taken:

  • ( P(E \cup S) = P(E) + P(S) - P(E \cap S) )

  • ( = 0.12 + 0.15 - 0.018 = 0.252 ) or 25.2%.

Additional Independent Events Example

• A student has a 20% chance of political affiliation and 30% for school leadership.

• The chance of either is 44%.

Calculations:

• Let A = Affiliated, L = Leadership:

• ( P(A) = 0.2 ), ( P(L) = 0.3 ), ( P(A \cup L) = 0.44 ).

• ( 0.44 = P(A) + P(L) - P(A \cap L) \rightarrow P(A \cap L) = 0.06 )

• Check independence:

• ( P(A \cap L) = P(A) \times P(L) = 0.2 \times 0.3 = 0.06 )

• Thus, A and L are independent.

Exhaustive Events

• A set of events that includes all possible outcomes of an experiment.

• At least one event must occur, encompassing the entire sample space.

• Example: Coin toss = heads and tails.

Mutually Exhaustive Events

• Events are both exhaustive and mutually exclusive.

• No two events can occur simultaneously.

• Example: Rolling a six-sided die: outcomes {1, 2, 3, 4, 5, 6} are mutually exhaustive.

Conditional Probabilities

• Formula:

  • ( P(A | B) = \frac{P(A \cap B)}{P(B)} )

  • This signifies the probability of A occurring given that B has occurred.

Summary

• Range of Values:

  • ( 0 \leq P(A) \leq 1 )

• Complements:

  • ( P(A^c) = 1 - P(A) )

• Mutually Exclusive Events:

  • ( P(A \cap B) = 0 )

• Independent Events:

  • ( P(A \cap B) = P(A) \times P(B) )

• Union:

  • ( P(A \cup B) = P(A) + P(B) - P(A \cap B) )

• Conditional Probabilities:

  • ( P(A | B) = \frac{P(A \cap B)}{P(B)} )

  • ( P(A \cap B) = P(A | B) \times P(B) .