1.2 — Probability Axioms and Rules#

Now that we understand sample spaces and events, we need a way to assign probabilities to events. This lesson introduces the probability axioms (fundamental rules that all probabilities must follow) and derived rules for calculating probabilities.

What is Probability?#

Probability is a number between 0 and 1 that quantifies how likely an event is to occur. Learn more: Probability

• 0: Impossible event (will never occur)
• 1: Certain event (will always occur)
• 0.5: Equally likely to occur or not occur

Notation: $P(A)$ denotes the probability of event $A$.

The Three Probability Axioms#

All probability theory is built on three fundamental axioms (rules that cannot be proven but are accepted as true). Learn more: Probability Axioms

Axiom 1: Non-Negativity#

For any event $A$, the probability is non-negative:

$P(A)\ge 0$

Interpretation: Probabilities cannot be negative.

Axiom 2: Normalization#

The probability of the sample space (certain event) is 1:

$P(\Omega )=1$

Interpretation: Something must happen. The sum of probabilities of all possible outcomes is 1.

Axiom 3: Additivity#

For any collection of mutually exclusive events $A_1,A_2,A_3,...$:

$P(A_1\cup A_2\cup A_3\cup ...)=P(A_1)+P(A_2)+P(A_3)+...$

Interpretation: The probability of “A or B or C…” equals the sum of individual probabilities when events cannot occur simultaneously.

Derived Probability Rules#

From these three axioms, we can derive many useful rules:

Rule 1: Complement Rule#

The probability of the complement of event $A$ is:

$P(A^c)=1-P(A)$

Proof: Since $A$ and $A^c$ are mutually exclusive and $A\cup A^c=\Omega$: $P(A)+P(A^c)=P(\Omega )=1$ Therefore: $P(A^c)=1-P(A)$

Example: If probability of rain is 0.3, then probability of no rain is $1-0.3=0.7$.

Rule 2: Probability of Impossible Event#

$P(\varnothing )=0$

Proof: Since ${\varnothing }^c=\Omega$ and $P(\Omega )=1$: $P(\varnothing )=1-P(\Omega )=1-1=0$

Rule 3: Monotonicity#

If event $A\subseteq B$, then:

$P(A)\le P(B)$

Interpretation: If $A$ always implies $B$, then $A$ cannot be more probable than $B$.

Example: If $A$ = “Roll a 6” and $B$ = “Roll an even number”, then $P(A)\le P(B)$.

Rule 4: Addition Rule (General Case)#

For any two events $A$ and $B$:

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

Why subtract? When we add $P(A)+P(B)$, we count outcomes in $A\cap B$ twice, so we subtract once.

Example:

• $A$ = “Roll an even number” = $\{2,4,6\}$, $P(A)=\frac{3}{6}=0.5$
• $B$ = “Roll a number > 4” = $\{5,6\}$, $P(B)=\frac{2}{6}=\frac{1}{3}$
• $A\cap B$ = $\{6\}$, $P(A\cap B)=\frac{1}{6}$
• $P(A\cup B)=0.5+\frac{1}{3}-\frac{1}{6}=\frac{2}{3}$

Rule 5: Addition Rule for Mutually Exclusive Events#

If events $A$ and $B$ are mutually exclusive ($A\cap B=\varnothing$), then:

$P(A\cup B)=P(A)+P(B)$

This is a special case of Rule 4 where $P(A\cap B)=0$.

Equally Likely Outcomes#

When all outcomes in a sample space are equally likely, we can use:

$P(A)=\frac{\text{Number of outcomes in }A}{\text{Total number of outcomes}}=\frac{|A|}{|\Omega |}$

Examples#

Example 1: Fair Die

• Sample space: $\Omega =\{1,2,3,4,5,6\}$ (6 equally likely outcomes)
• Event $A$ = “Roll a 6” = $\{6\}$
• $P(A)=\frac{1}{6}$

Example 2: Even Number

• Event $B$ = “Roll an even number” = $\{2,4,6\}$
• $P(B)=\frac{3}{6}=\frac{1}{2}$

Example 3: Card Draw

• Drawing one card from a standard 52-card deck
• Event $A$ = “Draw an ace” = 4 aces
• $P(A)=\frac{4}{52}=\frac{1}{13}$

Probability Bounds#

From the axioms, we can show that for any event $A$:

$0\le P(A)\le 1$

This follows from:

• Axiom 1: $P(A)\ge 0$
• Monotonicity: Since $A\subseteq \Omega$, we have $P(A)\le P(\Omega )=1$

Financial Applications#

Portfolio Returns#

Example: Portfolio can have three outcomes:

• High return: $P(\text{High})=0.3$
• Medium return: $P(\text{Medium})=0.5$
• Low return: $P(\text{Low})=0.2$

Check: $0.3+0.5+0.2=1.0$ ✓ (satisfies normalization)

Question: What’s the probability of NOT getting a low return?

• $P({\text{Low}}^c)=1-P(\text{Low})=1-0.2=0.8$

Risk Events#

Example: Two independent risk events:

• $A$ = “Market crash”: $P(A)=0.05$
• $B$ = “Liquidity crisis”: $P(B)=0.03$
• $A\cap B$ = “Both occur”: $P(A\cap B)=0.001$ (assumed)

Question: What’s the probability of at least one risk event?

• $P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.05+0.03-0.001=0.079$

Trading Outcomes#

Example: Daily trading outcomes:

• Profit: $P(\text{Profit})=0.6$
• Loss: $P(\text{Loss})=0.3$
• Break-even: $P(\text{Break-even})=0.1$

Check: $0.6+0.3+0.1=1.0$ ✓

Python Implementation#

Let’s implement probability calculations in Python:

def probability_complement(p_a):
    """Calculate probability of complement event"""
    return 1 - p_a

def probability_union(p_a, p_b, p_intersection):
    """Calculate P(A ∪ B) = P(A) + P(B) - P(A ∩ B)"""
    return p_a + p_b - p_intersection

def probability_union_mutually_exclusive(p_a, p_b):
    """Calculate P(A ∪ B) when A and B are mutually exclusive"""
    return p_a + p_b

# Example: Die roll
p_even = 3/6  # Probability of even number
p_greater_4 = 2/6  # Probability of > 4
p_six = 1/6  # Probability of 6 (intersection)

# Probability of even OR > 4
p_even_or_greater_4 = probability_union(p_even, p_greater_4, p_six)
print(f"P(even OR > 4) = {p_even_or_greater_4:.3f}")

# Probability of NOT even
p_not_even = probability_complement(p_even)
print(f"P(not even) = {p_not_even:.3f}")

# Example: Mutually exclusive events
p_one = 1/6  # Probability of rolling 1
p_six = 1/6  # Probability of rolling 6
p_one_or_six = probability_union_mutually_exclusive(p_one, p_six)
print(f"P(1 OR 6) = {p_one_or_six:.3f}")

Common Mistakes#

1. Adding probabilities without checking for overlap

   • Wrong: $P(A\cup B)=P(A)+P(B)$ (only true if mutually exclusive)
   • Right: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

2. Forgetting the complement rule

   • Instead of calculating $P(A^c)$ directly, use $1-P(A)$

3. Assuming equally likely outcomes

   • Only use $\frac{|A|}{|\Omega |}$ when outcomes are equally likely
   • In finance, outcomes are rarely equally likely!

Key Takeaways#

1. Three Axioms: Non-negativity, Normalization, Additivity
2. Complement Rule: $P(A^c)=1-P(A)$
3. Addition Rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
4. Mutually Exclusive: $P(A\cup B)=P(A)+P(B)$ when $A\cap B=\varnothing$
5. Equally Likely: $P(A)=\frac{|A|}{|\Omega |}$ when all outcomes are equally likely

Next Steps#

In the next lesson, we’ll explore independence and dependence between events, which is crucial for understanding relationships in financial data and risk analysis.