Probability - Formulas & Rules
IB Mathematics Analysis & Approaches (SL & HL)
🎲 Basic Probability
Theoretical Probability:
\[P(A) = \frac{n(A)}{n(U)}\]
where \(n(A)\) = number of favorable outcomes, \(n(U)\) = total outcomes
Given in formula booklet
Probability Range:
\[0 \leq P(A) \leq 1\]
Sum of All Probabilities:
\[\sum P(\text{all outcomes}) = 1\]
Expected Number of Occurrences:
\[\text{Expected} = n \times P(A)\]
where \(n\) = number of trials
🔄 Complementary Events
Complement Rule:
\[P(A') = 1 - P(A)\]
where \(A'\) means "NOT A" (complement of A)
Given in formula booklet
Alternative Form:
\[P(A) + P(A') = 1\]
➕ Addition Rule (OR)
General Addition Rule:
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Probability of A OR B (or both)
Given in formula booklet
Why We Subtract:
We subtract \(P(A \cap B)\) because events that satisfy both A and B are counted twice (once in \(P(A)\) and once in \(P(B)\))
⚡ Mutually Exclusive Events
Definition:
Events that cannot occur at the same time
Condition:
\[P(A \cap B) = 0\]
Addition Rule for Mutually Exclusive Events:
\[P(A \cup B) = P(A) + P(B)\]
Since they can't happen together, no need to subtract
✖️ Multiplication Rule (AND)
General Multiplication Rule:
\[P(A \cap B) = P(A) \times P(B|A)\]
Probability of A AND B occurring
Given in formula booklet
Alternative Form:
\[P(A \cap B) = P(B) \times P(A|B)\]
🔀 Conditional Probability
Formula:
\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]
Probability of A given that B has occurred
Given in formula booklet
Alternative Form:
\[P(B|A) = \frac{P(A \cap B)}{P(A)}\]
Important Note:
\(P(A|B)\) is generally NOT equal to \(P(B|A)\)
🔓 Independent Events
Definition:
Events where the occurrence of one does NOT affect the probability of the other
Test for Independence:
\[P(A \cap B) = P(A) \times P(B)\]
Equivalent Conditions:
• \(P(A|B) = P(A)\)
• \(P(B|A) = P(B)\)
🔄 Bayes' Theorem (HL)
Basic Form (Two Events):
\[P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}\]
Given in formula booklet
Extended Form (Two Mutually Exclusive Events):
\[P(A|B) = \frac{P(A) \times P(B|A)}{P(A) \times P(B|A) + P(A') \times P(B|A')}\]
Given in formula booklet
Extended Form (Three Mutually Exclusive Events):
\[P(A_1|B) = \frac{P(A_1) \times P(B|A_1)}{P(A_1) \times P(B|A_1) + P(A_2) \times P(B|A_2) + P(A_3) \times P(B|A_3)}\]
Given in formula booklet
📊 Discrete Probability Distributions
Requirements:
• \(P(X = x_i) \geq 0\) for all values
• \(\sum P(X = x_i) = 1\)
Expected Value (Mean):
\[E(X) = \mu = \sum x_i P(X = x_i)\]
Given in formula booklet
Variance:
\[\text{Var}(X) = E(X^2) - [E(X)]^2\]
\[\text{Var}(X) = \sum(x_i - \mu)^2 P(X = x_i)\]
Both forms given in formula booklet
🎯 Binomial Distribution
Notation:
\[X \sim B(n, p)\]
where \(n\) = number of trials, \(p\) = probability of success
Probability Formula:
\[P(X = r) = \binom{n}{r}p^r(1-p)^{n-r}\]
Given in formula booklet
Mean:
\[E(X) = np\]
Given in formula booklet
Variance:
\[\text{Var}(X) = np(1-p)\]
Also written as \(npq\) where \(q = 1-p\)
Given in formula booklet
⭕ Venn Diagrams
Key Regions:
• Intersection \(A \cap B\): Overlapping region (A AND B)
• Union \(A \cup B\): All of A and B combined (A OR B)
• A only: \(A \cap B'\) or \(A - B\)
• B only: \(B \cap A'\) or \(B - A\)
• Neither: \((A \cup B)'\)
Total Elements:
\[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]
🌲 Tree Diagrams
Purpose:
Visualize multi-stage experiments and conditional probabilities
Rules:
• Along branches: Multiply probabilities
• Between branches: Add probabilities
• Each complete branch represents a unique outcome
• Probabilities from each node must sum to 1
Calculating Probabilities:
To find the probability of a complete path:
Multiply all probabilities along that path
💡 Exam Tip: Most probability formulas are given in the IB formula booklet including addition rule, multiplication rule, conditional probability, Bayes' theorem, binomial distribution, and expected value. Always draw diagrams (Venn or tree) when solving probability problems - they help visualize the situation and prevent errors. Remember: multiply along branches, add between branches. For binomial: fixed trials, two outcomes, constant probability, independent. Use your GDC for binomial calculations!