Probability

Module 22 — Lesson 0022

Probability measures how likely an event is to occur — from 0 (impossible) to 1 (certain). In aptitude tests it often involves coins, dice, cards, and balls.

1. The Fundamental Formula

P(E) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)

Sample Space (S): the set of all possible outcomes.
Event (E): the specific outcome(s) we are interested in.
Complementary Event: P(not E) = 1 − P(E).

2. Common Sample Spaces

a) Coins

Coins	Sample Space	Total
1 coin	{H, T}	2
2 coins	{HH, HT, TH, TT}	4
3 coins	{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}	8

General rule: for n coins, total outcomes = 2ⁿ.

b) Dice

Dice	Sample Space	Total
1 die	{1, 2, 3, 4, 5, 6}	6
2 dice	(1,1), (1,2) … (6,6)	36

General rule: for n dice, total outcomes = 6ⁿ.

c) Playing Cards

Total cards: 52
Suits (4): Spades, Clubs (Black); Hearts, Diamonds (Red) — 13 each
Face cards (12): Jack, Queen, King (4 of each)
Aces (4): one per suit

3. Types of Events

a) Independent Events (AND Rule)

Events that do not affect each other (e.g., rolling a die and tossing a coin).

P(A and B) = P(A) × P(B)

b) Mutually Exclusive Events (OR Rule)

Events that cannot happen at the same time (e.g., drawing a red card OR a black card).

P(A or B) = P(A) + P(B)

c) Non-Mutually Exclusive Events

Events that can overlap (e.g., drawing a heart OR a face card — some cards are both).

P(A or B) = P(A) + P(B) − P(A and B)

4. Advanced Concepts

a) Selection without Replacement

When an item is picked and not put back, the total decreases for the next pick.

Example: probability of drawing 2 kings from a pack.
1st draw: 4/52.
2nd draw (only 3 kings left, 51 cards total): 3/51.
Total P = (4/52) × (3/51).

b) Bernoulli Trials (Binomial Probability)

Used when there are exactly two outcomes (success / failure) repeated n times.

P(r successes in n trials) = ⁿC_r × p^r × q^(n−r)

n: total trials
r: number of successes required
p: probability of success in one trial
q: probability of failure (1 − p)

c) "At Least One" Rule

Calculating "at least one" directly is often hard. It is easier to calculate "none" and subtract from 1.

P(At least one) = 1 − P(None)

5. Summary of Variables

Symbol	Meaning
`S`	Sample space
`E`	Event
`n(E)`	Number of favorable outcomes
`n(S)`	Total number of outcomes
`P(A)`	Probability of event A
`ⁿC_r`	Combination (selection)

For n coins, total outcomes are?

For n dice, total outcomes are?

Independent events A and B: P(A and B) = ?

Mutually exclusive A or B (cannot co-occur): P(A or B) = ?

Non-mutually exclusive (can overlap): P(A or B) = ?

Probability of drawing 2 kings without replacement?

Binomial: P(r successes in n trials) = ?

P(At least one) is most easily computed as?

Notes

Probability :- Basics :-

P(E) ⇒ favorable / total ; range 0→1

S ⇒ sample space (all outcomes) ; E ⇒ event ; P(not E) = 1 − P(E)

Sample Spaces :-

Coins ⇒ n coins → 2ⁿ (1→2, 2→4, 3→8) ; Dice ⇒ n dice → 6ⁿ (1→6, 2→36)

Cards ⇒ 52 total ; 4 suits ×13 ; face 12 (J,Q,K×4) ; aces 4

Event Types :-

a) Independent (AND) ⇒ do not affect each other → P(A and B) = P(A)×P(B)

b) Mutually exclusive (OR, no overlap) ⇒ P(A or B) = P(A)+P(B)

c) Non-mutually exclusive (overlap) ⇒ P(A or B) = P(A)+P(B) − P(A and B)

Advanced :-

Without replacement ⇒ totals shrink (eg 2 kings: 4/52 × 3/51)

Binomial ⇒ ⁿCr × pʳ × qⁿ⁻ʳ ; p = success, q = 1−p

At least one ⇒ 1 − P(None) (complement easier)

Variables :-

S ⇒ sample space ; n(S) ⇒ total outcomes ; n(E) ⇒ favorable

P(A) ⇒ prob of A ; ⁿCr ⇒ combinations

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