Probability Function & Classic Rules

An Infographic Guide (Using a Die)

Meet the Die: Our Random Experiment

Our journey into probability functions and rules will use a familiar friend: a standard, fair six-sided die.

When we roll this die, we are performing a random experiment – an action whose outcome is uncertain.

Key Concept: Sample Space (S)
The set of all possible outcomes of an experiment. For a single die roll, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Each outcome is equally likely if the die is fair.

The Probability Function (for a Fair Die)

A probability function (or probability mass function, PMF, for discrete outcomes like our die) assigns a probability to each possible outcome of our random variable X (the number rolled).

$$P(X = x)$$

This reads: "The probability that the random variable X takes the value x."

For a fair six-sided die:

$P(X = 1) = \frac{1}{6}$
$P(X = 2) = \frac{1}{6}$
$P(X = 3) = \frac{1}{6}$
$P(X = 4) = \frac{1}{6}$
$P(X = 5) = \frac{1}{6}$
$P(X = 6) = \frac{1}{6}$

(Each face has a $\frac{1}{6}$ chance)

Properties of a Probability Function:
1. The probability of any outcome x must be non-negative: $P(X = x) \ge 0$.
2. The sum of probabilities for all possible outcomes must equal 1: $\sum P(X = x) = 1$. (For our die, $6 \times \frac{1}{6} = 1$).

Rule 1: The Complement Rule

The complement of an event A (denoted A^c) is the event that A does NOT occur. The Complement Rule states:

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

"This means the probability of an event not happening is 1 minus the probability that it does happen."

Example: What is the probability of NOT rolling a 6?

Let A be the event of rolling a 6. So, $P(A) = P(X = 6) = \frac{1}{6}$.

The event A^c is "not rolling a 6".

$P(\text{not } 6) = 1 - P(6) = 1 - \frac{1}{6} = \frac{5}{6}$.

Key Concept: Complementary Events
An event and its complement A^c cover all possibilities in the sample space and are mutually exclusive (they cannot both happen).

Rule 2: Addition Rule (Mutually Exclusive Events)

If two events A and B are mutually exclusive (meaning they cannot occur at the same time), the probability that either A OR B occurs is the sum of their individual probabilities:

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

"If two things can't happen together, the chance of one OR the other happening is just their chances added up."

Example: What is the probability of rolling a 1 OR a 2?

Event A = rolling a 1, $P(A) = \frac{1}{6}$. Event B = rolling a 2, $P(B) = \frac{1}{6}$.

You cannot roll a 1 and a 2 at the same time, so they are mutually exclusive.

$P(1 \text{ or } 2) = P(1) + P(2) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$.

Key Concept: Mutually Exclusive Events
Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other. They have no outcomes in common. $P(A \cap B) = 0$.

Rule 3: General Addition Rule (Non-Mutually Exclusive)

If two events A and B are NOT mutually exclusive (they CAN occur at the same time), the probability that A OR B occurs is:

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

"We subtract $P(A \cap B)$ because we've double-counted the outcomes where both A and B happen."

Example: What is the probability of rolling an even number OR a number greater than 3?

Let $A = \text{rolling an even number} = \{2, 4, 6\}$. So $P(A) = \frac{3}{6}$.

Let $B = \text{rolling a number } > 3 = \{4, 5, 6\}$. So $P(B) = \frac{3}{6}$.

Events A and B can happen together: $\{4, 6\}$ are even AND > 3. So, $P(A \cap B) = \frac{2}{6}$.

$P(\text{Even} \cup >3) = P(A) + P(B) - P(A \cap B) = \frac{3}{6} + \frac{3}{6} - \frac{2}{6} = \frac{4}{6} = \frac{2}{3}$.

Visualizing the Outcomes:

$A$ = Even: { 2, 4, 6 }

$B$ = Greater than 3: { 4, 5, 6 }

$A \cap B$ (Overlap): { 4, 6 }

A: Even

B: >3

Key Concept: Intersection of Events ($A \cap B$)
The outcomes that are common to both event A and event B. If events are not mutually exclusive, their intersection is not empty.

Rule 4: Multiplication Rule (Independent Events)

If two events A and B are independent (the occurrence of one does not affect the probability of the other), the probability that BOTH A AND B occur is the product of their individual probabilities:

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

"If two things are unrelated, the chance of both happening is found by multiplying their individual chances."

Example: What is the probability of rolling a 6 on a first die roll AND rolling a 6 on a second die roll?

Let A = rolling a 6 on the first roll, $P(A) = \frac{1}{6}$.

Let B = rolling a 6 on the second roll, $P(B) = \frac{1}{6}$. These rolls are independent.

$P(\text{first is } 6 \cap \text{ second is } 6) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$.

Roll 1: $P(6) = \frac{1}{6}$

$ \times $

Roll 2: $P(6) = \frac{1}{6}$

Key Concept: Independent Events
Two events are independent if the outcome of one event does not influence the outcome of the other. Rolling a die multiple times typically involves independent events.