Probability 🎲

Classical definition, sample space, events — coins, dice, cards, aur word problems.

🌟 Probability Kya Hai?

🌦️ Weather Forecast wali example:

Kal baarish hogi ya nahi? Meteorologist kehta hai "70% chance of rain" — ye probability hai! Probability humein batati hai ki koi event kitni baar likely hai hona. Coin toss mein Head aane ki probability = 1/2 = 50% = 0.5.

Classical Probability — Basic Formula
P(Event E) = n(E) / n(S)

n(E) = Number of outcomes favourable to event E
n(S) = Total number of equally likely outcomes (Sample Space)
0 ≤ P(E) ≤ 1  (hamesha 0 aur 1 ke beech)
0
Impossible
0.25
Unlikely
0.5
Even chance
0.75
Likely
1
Certain
Important Properties:
• P(S) = 1 (certain event — sample space hamesha hoga)
• P(∅) = 0 (impossible event — empty set)
• P(Ē) = 1 − P(E) (complementary event)
• Sum of all probabilities = 1

🎰 Coin Problems

1 Coin: S = {H, T}, n(S) = 2

S = { Head (H), Tail (T) }

2 Coins: S = {HH, HT, TH, TT}, n(S) = 4

HH = (Head, Head)
HT = (Head, Tail)
TH = (Tail, Head)
TT = (Tail, Tail)

3 Coins: n(S) = 8

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
Example 1: 2 coins. P(exactly 1 head)?
Favourable: {HT, TH} → n(E) = 2
P = 2/4 = 1/2
Example 2: 3 coins. P(at least 2 heads)?
At least 2 heads: {HHH, HHT, HTH, THH} → n(E) = 4
P = 4/8 = 1/2
Example 3: 2 coins. P(ek bhi head nahi)?
No head: {TT} → n(E) = 1. P = 1/4
Ya: P(at least 1 head) = 3/4. P(no head) = 1 − 3/4 = 1/4

🎲 Dice Problems

1 Dice: n(S) = 6

1
2
3
4
5
6

🟢 Green = Prime numbers (2, 3, 5)

Example: 1 dice. P(prime), P(even), P(greater than 4)
P(prime) = P({2,3,5}) = 3/6 = 1/2
P(even) = P({2,4,6}) = 3/6 = 1/2
P(>4) = P({5,6}) = 2/6 = 1/3

2 Dice: n(S) = 36

Possible sums: 2 to 12. Sum=7 ke sabse zyada ways (6): (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)
Most likely sum = 7!
Sum23456789101112
Ways12345654321
P1/362/363/364/365/366/365/364/363/362/361/36
Example: 2 dice. P(sum=8), P(doubles), P(sum divisible by 4)
P(sum=8) = 5/36 [(2,6),(3,5),(4,4),(5,3),(6,2)]
P(doubles) = 6/36 = 1/6 [(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)]
P(div by 4) = P({4,8,12}) = (3+5+1)/36 = 9/36 = 1/4

♠♥♦♣ Card Problems

♠ Spades

13 cards
Black suit

♥ Hearts

13 cards
Red suit

♦ Diamonds

13 cards
Red suit

♣ Clubs

13 cards
Black suit

Total cards = 52. Each suit: Ace, 2,3,4,5,6,7,8,9,10, Jack, Queen, King
Face cards: J, Q, K = 3 × 4 = 12
Number cards: 2−10 = 9 × 4 = 36
Aces: 4 (one per suit)
Red cards = 26 (♥+♦), Black cards = 26 (♠+♣)
Examples from 52-card deck (1 card drawn randomly):
P(Ace) = 4/52 = 1/13
P(Face card) = 12/52 = 3/13
P(Red face card) = 6/52 = 3/26
P(King of black suit) = 2/52 = 1/26
P(Not a face card) = 40/52 = 10/13
Example: P(card ka number 5 se zyada ho)?
Numbers > 5: 6,7,8,9,10, J,Q,K → 8 types × 4 suits = 32 cards. But wait — also counting Ace as 1.
Cards > 5: {6,7,8,9,10,J,Q,K} = 8 × 4 = 32
P = 32/52 = 8/13

🎰 Bag / Ball Problems

Example: Bag mein 4 red, 6 blue, 5 green balls. 1 randomly nikalo.
n(S) = 15
P(Red) = 4/15
P(Blue or Green) = 11/15
P(Not Red) = 1 − 4/15 = 11/15
P(Yellow) = 0 (impossible!)
Example: k white aur 6 black balls. P(white) = 3/7. k nikalo.
k/(k+6) = 3/7 → 7k = 3k+18 → 4k = 18 → k = 4.5? — balls integer hote hain.
Aisa question mein usually integer aata hai. Let's try: P(white)=1/3 → k/(k+6)=1/3 → 3k=k+6 → k=3 balls ✓

🔢 Complementary Events

P(not E) = 1 − P(E)
Ye shortcut tab use karo jab P(E) nikalna mushkil ho
aur P(not E) calculate karna easy ho!

Example: P(at least one head in 3 coins)
= 1 − P(no heads) = 1 − 1/8 = 7/8
Example: 1000 lottery tickets mein se 5 winning hain. P(ticket win nahi karega)?
P(win) = 5/1000 = 1/200
P(not win) = 1 − 1/200 = 199/200

📝 Practice Questions

Q1. 2 dice fenke. P(sum = 6 ya 7)?

Sum=6: (1,5),(2,4),(3,3),(4,2),(5,1) → 5. Sum=7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6. Total=11/36

Q2. Ek bag mein 5 white, 7 black, 4 red balls hain. P(white ya red)?

P(white or red) = (5+4)/16 = 9/16

Q3. 52 cards mein se 2 cards remove ki gayi (ek red, ek black). Remaining cards mein se P(red face card)?

Total remaining = 50. Red cards remaining = 26−1 = 25. Red face cards = 6 (unchanged unless removed card was a red face card — assume not). P = 6/50 = 3/25

Q4. 1 se 100 tak ek number randomly select karo. P(perfect square)?

Perfect squares: 1,4,9,16,25,36,49,64,81,100 = 10 numbers. P = 10/100 = 1/10

Q5. Ek box mein kuch cards hain jo 6 se 50 tak ke multiples of 3 par likhe hain. Ek card randomly nikalo. P(number 50 se chhota aur even ho)?

Multiples of 3 from 6 to 50: 6,9,12,15,18,21,24,27,30,33,36,39,42,45,48 = 15 numbers.
Even multiples of 3 (multiples of 6) less than 50: 6,12,18,24,30,36,42,48 = 8 numbers.
P = 8/15
🎮 Probability Lab →