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Numerical Ability · AFCAT

NA12 — Probability

🎲 Numerical Ability – NA12 AFCAT Level ☆ Low Priority

📌 AFCAT Focus 2022–2026: Probability is low-weightage (0–1 question). Questions are standard: coins, dice, or cards. Memorise the sample spaces (coin=2, 2coins=4, dice=6, cards=52) and the three standard formulas. Questions are always simple substitution.

Topic A Definitions & Types of Events AFCAT Theory

Experiment

Any action with a well-defined set of outcomes. E.g., tossing a coin, rolling a die, drawing a card.

Sample Space (S)

The set of ALL possible outcomes. Tossing 2 coins: S = {HH, HT, TH, TT}, n(S) = 4.

Event (E)

Any subset of the sample space. "Getting at least 1 head" on 2 coins = {HH, HT, TH} = 3 outcomes.

Impossible Event

Event that can NEVER occur. P = 0. E.g., getting a 7 on a standard die.

Certain Event

Event that ALWAYS occurs. P = 1. E.g., getting a number ≤ 6 on a standard die.

Complementary

P(Ē) = 1 − P(E). Event "not E". E.g., P(not getting a 6) = 1 − 1/6 = 5/6. Use for "at least 1" problems.

Mutually Exclusive

Two events that cannot happen simultaneously. E.g., Head AND Tail on one toss. P(A or B) = P(A) + P(B).

Independent Events

Occurrence of one does not affect the other. E.g., tossing two different coins. P(A and B) = P(A) × P(B).

Topic B Balls from a Bag AFCAT Direct

Setup

A bag has r red, b blue, g green balls. Total = r+b+g. Drawing is random (equally likely).

P(specific colour)

P(red) = r/(r+b+g). P(not red) = (b+g)/(r+b+g). Numerator = favourable, denominator = total.

With Replacement

Ball is put back before next draw. Total stays same. Draws are independent. P stays same each time.

Without Replacement

Ball is NOT put back. Total reduces by 1 each draw. Draws are dependent. P changes after each draw.
E.g., bag: 3 red, 2 blue. P(2nd red | 1st was red) = 2/4 = 1/2 (not 3/5).

Example

Bag: 5 red, 3 blue, 2 green (total=10). P(blue) = 3/10. P(NOT green) = 8/10 = 4/5. P(red or blue) = 8/10 = 4/5.

1. Probability Basics

Fig 1.1 — Sample Spaces: Coins, Dice & Cards

P(E) = Favourable Outcomes ÷ Total Outcomes

🪙 COINS

1 coin → n = 2
{H, T}
2 coins → n = 4
{HH, HT, TH, TT}
3 coins → n = 8
P(all H) = 1/8
n coins → 2ⁿ outcomes

🎲 DICE

1 die → n = 6
{1, 2, 3, 4, 5, 6}
P(even) = 3/6 = 1/2
Evens: 2, 4, 6
P(prime) = 3/6 = 1/2
Primes: 2, 3, 5
2 dice → n = 36
Sum=7: P = 6/36 = 1/6

🃏 CARDS (52)

4 suits × 13 = 52
♠ ♥ ♦ ♣ (13 each)
P(Ace) = 4/52 = 1/13
P(Heart) = 13/52 = 1/4
Face cards (J,Q,K) = 12
P(face) = 12/52 = 3/13
Red cards (♥+♦) = 26
P(red) = 26/52 = 1/2

🎲 Key Probability Formulas

P(E) = n(E)/n(S) where n(S) = total outcomes
0 ≤ P(E) ≤ 1 always
P(Ē) = 1 − P(E) (complementary)
P(A or B) = P(A) + P(B) − P(A and B)
Mutually exclusive: P(A or B) = P(A) + P(B)

🎲 Quick Reference

Certain event: P = 1 (must happen)
Impossible event: P = 0 (cannot happen)
Independent events: P(A and B) = P(A) × P(B)
"At least one" = 1 − P(none)
With replacement → same P each draw

📐 Formula Sheet — NA12

Core Formula

P(E) = n(E) / n(S)
P(Ē) = 1 − P(E)
0 ≤ P ≤ 1
P(certain) = 1 | P(impossible) = 0

Sample Spaces

1 coin: 2 | 2 coins: 4 | 3 coins: 8
n coins: 2ⁿ
1 die: 6 | 2 dice: 36
Cards: 52 (4 suits × 13 ranks)

Cards Reference

Aces: 4 | Face cards (J,Q,K): 12
Each suit: 13 cards
Red (♥,♦): 26 | Black (♠,♣): 26
P(Ace) = 1/13 | P(Heart) = 1/4

Dice Reference

P(even) = 1/2 | P(prime 2,3,5) = 1/2
2 dice sum = 7: P = 1/6 (most likely)
2 dice sum = 2 or 12: P = 1/36
2 dice sum = 6: 5 ways → 5/36

📝 Topic-Wise PYQs — NA12

Q1. A card is drawn from a pack of 52. Probability of getting a King is: AFCAT PYQ

(a) 1/13(b) 1/52(c) 4/13(d) 1/26

✔ Answer: (a) 1/13

4 Kings in 52 cards. P = 4/52 = 1/13.

Q2. Two coins are tossed. Probability of getting exactly one head is: AFCAT PYQ

(a) 1/2(b) 1/4(c) 3/4(d) 1

✔ Answer: (a) 1/2

Sample space = {HH, HT, TH, TT}. Exactly 1 head = {HT, TH} = 2 outcomes. P = 2/4 = 1/2.

Q3. A die is rolled. Probability of getting a prime number is: AFCAT PYQ

(a) 1/3(b) 1/2(c) 2/3(d) 1/6

✔ Answer: (b) 1/2

Primes on a die: 2, 3, 5 → 3 outcomes. P = 3/6 = 1/2.

Q4. Probability of drawing a red face card from 52 cards is: ⚡ Tricky

(a) 3/26(b) 1/13(c) 6/52(d) Both a & c

✔ Answer: (d) Both a & c

Red face cards: J, Q, K of Hearts + J, Q, K of Diamonds = 6 cards. P = 6/52 = 3/26. Both (a) and (c) are equivalent.

Q5. A bag has 4 red, 3 blue, 2 green balls. P(not green) = ? AFCAT PYQ

(a) 2/9(b) 7/9(c) 4/9(d) 1/9

✔ Answer: (b) 7/9

Total = 9. Green = 2. P(not green) = 1 − 2/9 = 7/9.

Q6. Two cards drawn without replacement from 52. P(both aces) = ? ⚡ Tricky

(a) 1/169(b) 1/221(c) 4/52(d) 1/26

✔ Answer: (b) 1/221

P(1st ace) = 4/52. P(2nd ace | 1st was ace) = 3/51. P(both) = 4/52 × 3/51 = 12/2652 = 1/221. Key: "without replacement" means denominator reduces to 51.

🧠 Quick Memory Chart — NA12

🎲 Core

P(E) = n(E)/n(S)
P(Ē) = 1 − P(E)
0 ≤ P ≤ 1
At least 1 = 1 − P(none)

🎲 Spaces

1 coin: 2
2 coins: 4 | 3 coins: 8
1 die: 6 | 2 dice: 36
Cards: 52

🃏 Cards

Aces: 4
Face cards (J,Q,K): 12
Red / Black: 26 each
P(King) = 1/13

📝 Mock Tests 🎯 Subject Quiz ✈️ Telegram