Probability

1. Definition

• Probability = a number that expresses how likely an event is to happen.
• It ranges from 0 to 1:
  • 0 → impossible (event will never happen).
  • 1 → certain (event will always happen).
  • Numbers in between represent varying degrees of likelihood.

Example: Probability of flipping a coin and getting heads = 0.5 (50%).

---

2. Basic Formula

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

• $P(E)$ = probability of event $E$.

Example: Rolling a die.

• Event = rolling a 4.
• Favorable outcomes = 1 (just the “4”).
• Total outcomes = 6.

$P(\text{rolling a 4}) = \frac{1}{6} \approx 0.167$

---

3. Key Types of Probability

• Theoretical probability → based on math (fair dice, fair coin).
• Experimental probability → based on actual experiments (flip a coin 100 times).
• Subjective probability → based on belief or experience (chance of rain tomorrow).

---

4. Common Rules

1. Complement Rule

$P(\text{not A}) = 1 – P(A)$

If probability of rain is 0.3, probability of no rain = 0.7.

2. Addition Rule (OR)
   For mutually exclusive events:

$P(A \text{ or } B) = P(A) + P(B)$

3. Multiplication Rule (AND)
   For independent events:

$P(A \text{ and } B) = P(A) \times P(B)$

Example: Rolling a 2 on one die and a 5 on another:

$\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

---

5. Everyday Examples

• Weather forecast → “70% chance of rain.”
• Sports → “Team has 0.8 probability of winning.”
• Medicine → “Drug has a 5% probability of side effects.”

---

Summary:
Probability = a measure of how likely an event is, between 0 (impossible) and 1 (certain).
It’s calculated as favorable outcomes ÷ total outcomes, and follows simple rules (complement, addition, multiplication).