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Counting and Probability - Coggle Diagram
Counting and Probability
Language of Events and Sets
Set Operations in Probability
complement of A: A'
Union: A∪B (either or both events occurring)
Three-way operations: A∩B∩C and A∪B∪C
Intersection: A∩B (events occurring together)
Mutually Exclusive Events
examples: Rolling an odd and even number simultaneously.
Identifying events that cannot occur together
Fundamental Concepts
Events
An event is a subset of the sample space that consists of one or more outcomes.
Simple Event: Contains only one outcome (e.g., rolling a 4 on a die).
Compound Event: Contains multiple outcomes (e.g., rolling an even number).
Certain Event: Contains all outcomes (e.g., rolling a number between 1 and 6).
Impossible Event: Contains no outcomes (e.g., rolling a 7 on a standard die).
Sample Spaces (See Annex 1)
A sample space is the set of all possible outcomes for a given experiment
Rolling a six-sided die
S={1,2,3,4,5,6}
Finite Sample Space: Contains a limited number of outcomes (e.g., rolling a die).
Infinite Sample Space: Contains infinitely many outcomes (e.g., measuring a person’s height).
Discrete Sample Space: Outcomes are countable (e.g., number of heads in a coin toss).
Continuous Sample Space: Outcomes are uncountable and fall within an interval (e.g., time taken to finish a race).
Outcomes
An outcome is a single possible result of a probability experiment.
Rolling a die → Possible outcomes: 1, 2, 3, 4, 5, 6
Real-World Applications
Using everyday scenarios
rolling dice
choosing cards
Combinations
Combination Formula (See Annex 2)
r! (r factorial) = Product of all positive integers from 1 to r.
(n−r)! accounts for the ignored order in combinations.
n! (n factorial) = Product of all positive integers from 1 to n.
Pascal’s Triangle & Binomial Coefficients
Exploring Pascal’s Triangle patterns (See Annex 3)
Connecting Pascal’s Triangle to binomial expansions ((x+y)n
Row 4: 1, 4, 6, 4, 1
(x+y)4
(x+y)4 = C(4,0)x4 +C(4,1)x3y +C(4,2)x2y2+ C(4,3)xy3+ C(4,4)y4
1x4+ 4x3y+ 6x2y2+ 4xy3+ 1y4
Pascal’s Triangle is a powerful visual representation of combinations and binomial coefficients, providing a quick way to expand expressions and solve counting problems without using factorial calculations.
Understanding Combinations
Difference between permutations and combinations
permutations
Arrangements of objects where order is important.
Link Title
Application:
Seating arrangements
Ranking in a race
Creating passwords
Positioning matters; changing the order creates a different arrangement.
combinations
Positioning does not matter; selecting the same objects in different orders is counted as one.
Application:
Lottery draws
Forming teams
Selecting committee members
Selections of objects where order is not important.
Definition
A combination is a selection of r objects from a set of n distinct objects without considering the order of selection.
Fundamentals of Probability
Probability Rules
Complement Rule: P(A') = 1 - P(A)
Addition Rule: P(A∪B)=P(A)+P(B)−P(A∩B)
Empirical Probability (Relative Frequency)
Estimating probabilities using data
Example: probability of an event occurring in repeated trials
Probability as a Measure of Likelihood
Definition: 0≤P(A)≤1
P(A) = 0 → Impossible Event (e.g., rolling a 7 on a standard die)
P(A) = 1 → Certain Event (e.g., rolling a number between 1 and 6 on a die)
0 < P(A) < 1 → Event is possible but not certain (e.g., rolling an even number)
Certainty (P(A)=1P(A)=1) vs. Impossibility (P(A)=0P(A)=0)
If P(A) is close to 1, the event is very likely to happen. If P(A) is close to 0, the event is very unlikely.
Conditional Probability and Independence (See Annex 4)
Conditional Probability Formula
Solving conditional probability problems
Formula: P(A∩B)=P(A∣B)P(B)
Independent vs. Dependent Events
Definition of Independence: P(A∣B)=P(A)
Independence Formula: P(A∩B)=P(A)P(B)
Understanding Conditional Probability
Definition of P(A∣B) (probability of A given B has occurred
Using Data to Check for Independence
Relative frequencies as an indication of independence
Examples using two-way tables
