Probability - Class 9 Maths

🎲 Probability Kya Hai?

Probability (Sambhavna) ye measure karti hai ki koi event (ghatna) kitni baar hone ki sambhavana (chance) hai. Kya aaj baarish hogi? Kya dice par 6 aayega? Kya coin par heads aayega? — Ye sab probability se samajh sakte hain.

🌟 Probability Hamesha 0 aur 1 ke beech hoti hai:

P = 0: Event KABHI nahi hoga. (Jaise ek normal coin par 3 aana)
P = 1: Event HAMESHA hoga. (Jaise sun ka east mein ugna)
0 < P < 1: Kuch chance hai. (Jaise rain today, heads in coin flip)

1️⃣ Important Terms (Zaruri Shabda)

Term	Matlab (Roman Hindi)	Example
Experiment	Koi action jo uncertain result de	Coin flip karna, dice roll karna
Outcome	Experiment ka ek possible result	Coin flip → Heads ya Tails
Sample Space (S)	Sabhi possible outcomes ka set	Coin: S = {H, T}; Dice: S = {1,2,3,4,5,6}
Event (E)	Outcomes ka ek group jiske liye P nikalte hain	Dice par even number aana: E = {2,4,6}
Favourable Outcomes	Woh outcomes jo event mein hain	Event "6 aana" ke liye: 1 favourable outcome

2️⃣ Theoretical Probability (Siddhaantik Sambhavna)

Theoretical probability tab use hoti hai jab hum pehle se jaante hain ki sabhi outcomes equally likely hain (jaise ek fair coin ya fair dice).

P(Event E) = \(\frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Outcomes in Sample Space}}\)
\( P(E) = \frac{n(E)}{n(S)} \)

📌 Important Properties:

\( 0 \leq P(E) \leq 1 \) hamesha
\( P(\text{Sure Event}) = 1 \)
\( P(\text{Impossible Event}) = 0 \)
\( P(E) + P(\bar{E}) = 1 \) → \( P(\bar{E}) = 1 - P(E) \) (Complementary Events)

📝 Example 1 — Coin Flip:

Ek fair coin flip karte hain. S = {H, T}. n(S) = 2.

P(Heads) = \(\frac{1}{2} = 0.5\) | P(Tails) = \(\frac{1}{2} = 0.5\) ✅

📝 Example 2 — Dice Roll:

Ek fair dice roll karte hain. S = {1, 2, 3, 4, 5, 6}. n(S) = 6.

P(3 aana) = \(\frac{1}{6}\)
P(Even number aana) = \(\frac{n(\{2,4,6\})}{6} = \frac{3}{6} = \frac{1}{2}\)
P(7 aana) = \(\frac{0}{6} = 0\) (impossible!)
P(6 se chhhota ya barabar) = \(\frac{6}{6} = 1\) (certain!)

📝 Example 3 — Playing Cards:

52 cards ki deck mein se ek card random nikala. P(Ace aana)?

4 Aces hain (♠♥♦♣). n(S) = 52.
P(Ace) = \(\frac{4}{52} = \frac{1}{13} \approx 0.077\) ✅

3️⃣ Empirical Probability (Anubhav se Milne wali Sambhavna)

Empirical (ya Experimental) probability tab use hoti hai jab hum actual experiment karte hain aur results observe karte hain. Theory se nahi, asli data se nikalte hain.

Empirical P(E) = \(\frac{\text{Number of times E occurred}}{\text{Total number of trials}}\)

🔍 Empirical vs Theoretical:

Ek coin ko 100 baar flip karo. Theory kehta hai P(H) = 0.5 = 50 baar. Par actual experiment mein tum paa sakte ho 48 ya 53 baar Heads — ye empirical probability hai. Jitne zyada trials karoge, empirical → theoretical ke paas aata jaayega. Ise Law of Large Numbers kehte hain!

📝 Example — Bag of Balls:

Ek bag mein red aur blue balls hain. 50 baar ek ball nikali aur wapas rakhi. Red ball 30 baar aayi.

Empirical P(Red) = \(\frac{30}{50} = \frac{3}{5} = 0.6\)
Iska matlab bag mein approximately 60% balls red hain.

4️⃣ Tree Diagrams (Ped wale Diagram)

Jab ek se zyada events hon (compound events), tab Tree Diagram sabhi outcomes systematically list karne mein help karta hai.

Sample Space S = {HH, HT, TH, TT} → 4 outcomes.

P(exactly 1 Heads) = \(\frac{2}{4} = \frac{1}{2}\) (HT aur TH dono count hote hain)

P(2 Heads) = \(\frac{1}{4}\) | P(no Heads) = \(\frac{1}{4}\)

📝 Example — Dice + Coin:

Ek dice roll karo aur coin flip karo. Kitne total outcomes hain? P(6 aur Heads)?

Dice: 6 outcomes × Coin: 2 outcomes = 12 total outcomes
Favourable (6 AND H): sirf 1 → P = \(\frac{1}{12}\) ✅

5️⃣ Complementary Events (Viparit Ghatna)

Kisi event E ka complement hota hai "E nahi hona" — ise \(\bar{E}\) ya E' likhte hain.

P(E) + P(\(\bar{E}\)) = 1 → P(\(\bar{E}\)) = 1 - P(E)

📝 Example:

Dice par 6 na aane ki probability kya hai?

P(6 aana) = 1/6
P(6 na aana) = 1 - 1/6 = 5/6 ✅

🎯 Practice Questions

❓ Q1:

Ek bag mein 3 red, 4 blue, 5 green balls hain. Ek ball random nikali. P(Blue) aur P(Not Blue) nikalo.

Answer: n(S) = 12
P(Blue) = \(\frac{4}{12} = \frac{1}{3}\)
P(Not Blue) = \(1 - \frac{1}{3} = \frac{2}{3}\) ✅

❓ Q2:

Ek fair dice do baar roll kiya. P(dono baar same number aaye)?

Answer: Total outcomes = 6×6 = 36
Same outcomes: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) = 6
P = \(\frac{6}{36} = \frac{1}{6}\) ✅

❓ Q3 (Challenge):

Ek experiment mein coin 200 baar flip ki. Heads 112 baar aaya. Empirical P(Heads) kya hai? Theoretical se kitna alag hai?

Answer:
Empirical P(H) = \(\frac{112}{200} = 0.56\)
Theoretical P(H) = 0.5
Difference = 0.56 - 0.50 = 0.06 — ye expected hai! Aur zyada trials se ye kam hota jaayega. 📉

🎲 Probability (Sambhavna)