1. Basic concepts
Probability measures how likely an event is. For any event E, 0 ≤ P(E) ≤ 1.
If sample space S has equally likely outcomes, P(E)=number of favorable outcomes / total outcomes
2. Complement
P(E') = 1 − P(E)
3. Addition rule
For events A and B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive, P(A ∩ B)=0 so P(A ∪ B)=P(A)+P(B)
4. Conditional probability & multiplication rule
P(A|B) = P(A ∩ B) / P(B)
P(A ∩ B) = P(A|B) × P(B)
If A and B are independent, P(A ∩ B)=P(A)P(B).
5. Examples
Example 1: Toss a fair coin twice. Probability of exactly one head?
Outcomes: HH, HT, TH, TT → favorable: HT, TH → P=2/4=1/2.
Example 2: Draw one card from a 52-card deck. P(ace or heart) = P(ace)+P(heart)-P(ace of hearts) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
Example 3 (conditional): Box has 3 red and 2 blue balls. Draw 2 without replacement. P(second red | first red)=?
After first red, remaining red=2, total=4 ⇒ 2/4=1/2.
6. Practice
1) Roll a fair die. P(odd number) = 3/6 = 1/2.
2) From 5 students, choose 2. What is P(that two chosen are both girls) if class has 3 girls and 2 boys? → C(3,2)/C(5,2)=3/10.