Probability: Concepts, Formulas, and Clear Examples

Probability $P(E)=\frac{\text{favorable}}{\text{total}}$ when outcomes are equally likely. Complement: $P(E')=1-P(E)$.

Campus MCQs use dice/cards/coins, conditional probability, and at least one via complement.

Independent events multiply; mutually exclusive events add (no overlap).

Classical definition

Equally likely

$$P(E) = \frac{n(E)}{n(S)}$$

Complement and OR

Complement

$$P(E') = 1 - P(E)$$

Mutually exclusive

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

Example — At least one head (2 coins)

$1 - P(\text{no head}) = 1 - \frac{1}{4} = \frac{3}{4}$.

Independent events

AND independent

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

Conditional

Conditional

$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$

Cards and dice counts

• Deck 52; each suit 13. Two dice: 36 outcomes.
• Face cards 12 (J,Q,K × 4 suits).

Bayes (light)

• Reverse conditional rare in aptitude; stick to definition and tree for two-step.

Step-by-step method for exam questions

1. Define sample space; count total equally likely outcomes.
2. Count favorable (or use complement).
3. Simplify fraction.
4. For two-stage, multiply along branches if independent.
5. Check $P\le1$.
6. Pick option.

Formula cheat sheet

Frequently asked questions

With replacement vs without?

Without: second draw denominator decreases; not independent.

Are A and A' independent?

No unless P(A)=0 or 1.

Add probabilities when?

Only if mutually exclusive.

Dice sum 7?

6 favorable of 36 → 1/6.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.