1. Introduction to Probability
Probability is the measure of the likelihood of an event occurring.
Sample space (S): All possible outcomes of an experiment.
Event (E): A subset of the sample space.
Example:
Tossing a die: S = {1, 2, 3, 4, 5, 6}
Event A (even numbers) = {2, 4, 6}
2. Types of Events
Mutually Exclusive Events: Two events that cannot occur together.
Example: Rolling even (A) or odd (B) numbers → A ∩ B = ∅
Non-Mutually Exclusive Events: Events that can occur together.
Example: Rolling even (A) or prime (C) numbers → A ∩ C = {2}
3. Probability Formulas
Probability of an event: P(E) = n(E) / n(S)
Addition Law:
For mutually exclusive events: P(A ∪ B) = P(A) + P(B)
For non-mutually exclusive events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Example:
Dice: P(even or prime) = 3/6 + 3/6 − 1/6 = 5/6
4. Independent and Dependent Events
Independent: Occurrence of one event does not affect the other.
Example: Tossing a coin and rolling a die → P(H ∩ 4) = 1/2 × 1/6 = 1/12
Dependent: Occurrence of one event affects the other.
Example: Drawing two cards without replacement → P(K ∩ A) = 4/52 × 4/51 = 4/663
5. Multiplication Principle
For independent events: P(A ∩ B) = P(A) × P(B)
Sample space when tossing a coin and rolling a die:
S = {(H,1),(H,2),…,(T,6)}, n(S) = 12
6. Probability with Cards
Standard deck: 52 cards, 4 suits, 13 cards each.
Examples:
King, Queen, or Jack: P(K ∪ Q ∪ J) = 1/13 + 1/13 + 1/13 = 3/13
Drawing two cards (with replacement): P(K ∩ A) = 1/13 × 1/13 = 1/169
Drawing two cards (without replacement): P(K ∩ A) = 4/52 × 4/51 = 4/663
7. Probability with Balls
Bag with 5 red, 6 blue balls:
With replacement:
P(R ∩ B) = 5/11 × 6/11 = 30/121
Without replacement:
P(R ∩ B) = 5/11 × 6/10 = 3/11
8. Tree Diagram
Visual representation of outcomes and probabilities.
Example 1: Tossing a coin twice → S = {HH, HT, TH, TT}
P(at least 2 heads) = 1/4
Example 2: Coin + die → P(H ∩ 4) = 1/12
9. Frequency Distribution & Measures of Central Tendency
Frequency Table: Organizes raw data into classes.
Mean: Average value
Median: Middle value
Mode: Most frequent value
Range: Max − Min
Quartiles: Divide data into 4 equal parts
Example: Heights of 50 people → Construct frequency table, find mean and median
10. Important Formulas Recap
P(E) = n(E)/n(S)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∩ B) = P(A) × P(B) (for independent events)
5 Most Important & Common Questions
1. Probability of getting an even or prime number on a die
Sample space: {1,2,3,4,5,6}
Even (A) = {2,4,6}, P(A) = 3/6 = 1/2
Prime (C) = {2,3,5}, P(C) = 3/6 = 1/2
Intersection: {2}, P(A ∩ C) = 1/6
P(A ∪ C) = 1/2 + 1/2 − 1/6 = 5/6
2. Probability of getting a king, queen, or jack from a deck
Total cards = 52
P(K) = P(Q) = P(J) = 4/52 = 1/13
P(K ∪ Q ∪ J) = 1/13 + 1/13 + 1/13 = 3/13
3. Probability of drawing two cards
With replacement: P(K ∩ A) = 1/13 × 1/13 = 1/169
Without replacement: P(K ∩ A) = 1/13 × 4/51 = 4/663
4. Probability using tree diagram (coin + die)
P(H ∩ 4) = 1/2 × 1/6 = 1/12
Tree diagram:
H → 1,2,3,4,5,6
T → 1,2,3,4,5,6
5. Probability of drawing two balls from a bag
Bag: 5 red, 6 blue
With replacement: P(R ∩ B) = 5/11 × 6/11 = 30/121
Without replacement: P(R ∩ B) = 5/11 × 6/10 = 3/11
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