Probability & Permutation/Combination

Section A · Futures First International — Market Trading Intern OA

📌 What Has Been Asked (OA & Rounds)

Format	Typical Question Style
OA (MCQ)	"Probability of getting exactly 2 heads in 5 coin tosses"
OA	"In how many ways can 5 people sit around a circular table?"
OA	"A bag has 4 red, 5 blue balls. 2 drawn at random. Probability both are red?"
OA	"How many 4-digit numbers can be formed using {1,2,3,4,5} without repetition?"
Buzzer	"Probability of at least one head in 3 tosses?" → 1 − (1/2)³ = 7/8
Interview	"I offer you a game: flip a coin — heads you win ₹150, tails you lose ₹100. Do you play?" → Expected Value question
Interview	"Given at least one child is a boy, probability both are boys?" → conditional probability classic

⚠️ Probability & EV are disproportionately important for trading interviews. Even if you see fewer in the OA, the interview rounds hammer this.

🧠 Core Concepts

1. Permutations (Order Matters)

nPr = n! / (n−r)!

n items in a line: n!
n items in a circle: (n−1)!
n items with repetitions (p alike, q alike): n! / (p! × q!)

2. Combinations (Order Doesn't Matter)

nCr = n! / [r! × (n−r)!]

Key identities:
nCr = nC(n−r)
nC0 = nCn = 1
nC1 = n

Quick Values (Memorize)

nCr	Value
5C2	10
6C2	15
6C3	20
7C2	21
7C3	35
8C2	28
8C3	56
10C2	45
10C3	120
10C4	210

3. Probability Basics

P(E) = Favorable outcomes / Total outcomes

P(not E) = 1 − P(E)

P(A or B) = P(A) + P(B) − P(A and B)

P(A and B) = P(A) × P(B)    [if independent]
P(A and B) = P(A) × P(B|A)  [if dependent]

4. Conditional Probability

P(A|B) = P(A ∩ B) / P(B)

Classic: "Given at least one child is a boy, P(both boys)?"

Sample space for 2 children: {BB, BG, GB, GG}
"At least one boy" eliminates GG → {BB, BG, GB}
P(both boys | at least one boy) = 1/3

5. Expected Value (CRITICAL for Trading)

E(X) = Σ [outcome × probability of that outcome]

Decision rule: If E(X) > 0, take the bet.

Example: "Heads you win ₹150, tails you lose ₹100."

E = 0.5 × 150 + 0.5 × (−100) = 75 − 50 = +25

→ Yes, play. Positive EV.

⚡ Speed Tricks for Futures First

Trick 1: "At least one" = 1 − P(none)

"P(at least one head in 4 tosses)" = 1 − (1/2)⁴ = 1 − 1/16 = 15/16

Trick 2: Balls without replacement — use combinations

"4 red, 5 blue. Draw 2. P(both red)?"

= 4C2 / 9C2 = 6/36 = 1/6

Trick 3: Dice problems — memorize key probabilities

Two dice → 36 total outcomes.

Sum	Ways	P
2	1	1/36
3	2	2/36
7	6	6/36 = 1/6
12	1	1/36
≥ 10	3+2+1=6	1/6

Trick 4: Circular permutation

"5 people around a table" = (5−1)! = 4! = 24 "Necklace/bracelet (can flip)" = (n−1)!/2

Trick 5: Cards — key numbers

52 cards, 4 suits (13 each), 12 face cards, 4 aces.

P(drawing a heart) = 13/52 = 1/4
P(drawing a king) = 4/52 = 1/13
P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13

Trick 6: Quick EV thinking

"Pay ₹10 to roll a die. Win the face value in ₹."

E(winnings) = (1+2+3+4+5+6)/6 = 3.5
E(profit) = 3.5 − 10 = −6.5 → Don't play

🎯 What Can Be Asked (Variations)

"How many ways to distribute 5 identical balls into 3 distinct boxes?" → Stars and bars: C(5+3−1, 3−1) = C(7,2) = 21
"In a room of 6 people, how many handshakes?" → 6C2 = 15
"A committee of 3 is formed from 5 men and 4 women. At least 1 woman. How many ways?" → Total − all men = 9C3 − 5C3 = 84 − 10 = 74
"You have a biased coin (P(H)=0.6). Flip 3 times. P(exactly 2 heads)?" → C(3,2) × (0.6)² × (0.4)¹ = 3 × 0.36 × 0.4 = 0.432
"Roll two dice. What's the probability sum is a prime number?" → Prime sums: 2,3,5,7,11. Count ways: 1+2+4+6+2 = 15. P = 15/36 = 5/12
Interview classic: "I roll a die. If it's ≥4, I pay you the face value. If <4, you pay me ₹3. Do you play?" → E = P(4)×4 + P(5)×5 + P(6)×6 − P(1)×3 − P(2)×3 − P(3)×3 → = (4+5+6)/6 − 3×3/6 = 15/6 − 9/6 = 6/6 = +1 → Play (positive EV)

📝 Practice Questions (with Answers)

Q1. In how many ways can the letters of the word "MISSISSIPPI" be arranged?

A: 11! / (4! × 4! × 2!) = 11! / (24 × 24 × 2) = 39916800/1152 = 34,650

Q2. A bag has 6 white and 4 black balls. 3 drawn randomly. P(all white)?

A: 6C3 / 10C3 = 20/120 = 1/6

Q3. P(getting a sum of 9 with two dice)?

A: Favorable: (3,6),(4,5),(5,4),(6,3) = 4 ways. P = 4/36 = 1/9

Q4. How many 3-digit numbers can be formed using {1,2,3,4,5} with repetition allowed?

A: 5 × 5 × 5 = 125

Q5. You flip a fair coin until you get heads. Expected number of flips?

A: Geometric distribution: E = 1/p = 1/(1/2) = 2 flips

Q6. "I'll pay you ₹X if a 6 appears on a die roll. You pay me ₹10 if it doesn't. What's the minimum X for you to accept?"

A: EV ≥ 0: (1/6)X − (5/6)(10) ≥ 0 → X ≥ 50. Minimum X = ₹50

Q7. 5 boys and 3 girls sit in a row. P(no two girls adjacent)?

A: Arrange 5 boys: 5! = 120. Girls choose 3 of 6 gaps: 6C3 = 20. Girls arrange: 3! = 6. Total favorable = 120 × 20 × 6 = 14,400. Total = 8! = 40,320. P = 14400/40320 = 5/14

🔑 Exam Strategy

EV questions are the most "trading-relevant" topic — interviewers are testing if you think like a trader
For combinatorics in OA: know nCr values for small n by heart (table above) — saves 20+ seconds
"At least one" = complement → always faster
For dice/cards: memorize the total outcomes and common favorable counts
In interviews, always explain your reasoning — "I'd compute the expected value: probability times payoff for each outcome..." — even if you make an arithmetic mistake, the framework scores points
P&C: if order matters → permutation; if not → combination. Ask yourself: "Does rearranging the selection give a different outcome?"

08_probability_pnc.md