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) .