Probability basics — sample space, events, P(E) = favourable - total

1.3.6 · Maths › Basic Data & Probability

Overview

Probability measure karta hai ki koi event kitni likely hai occur hone ki. Yeh uncertain situations mein outcomes predict karne ki foundation hai—coin flips se lekar weather forecasts tak, medical diagnoses tak.

1. Sample Space

Sample Spaces ke Examples

Experiment	Sample Space	Number of outcomes
Toss 1 coin	$S = {H, T}$	2
Roll1 die	$S = {1, 2, 3, 4, 5, 6}$	6
Toss 2 coins	$S = {HH, HT, TH, T}$	4
Pick a card	$S = {52 different cards}$	52

2. Events

Example Events

Ek single die roll karne ke liye jahan $S = {1, 2, 3, 4, 5, 6}$ :

Event Description	Event Set	Type
"Roll a 4"	$E_{1} = {4}$	Simple
"Roll an even number"	$E_{2} = {2, 4, 6}$	Compound
"Roll less than 7"	$E_{3} = {1, 2, 3, 4, 5, 6} = S$	Sure event
"Roll a 7"	$E_{4} = \emptyset$	Impossible

Figure — Probability basics — sample space, events, P(E) = favourable - total

3. The Fundamental Probability Formula

Yeh Formula Kyu Kaam Karta Hai — First Principles se Derivation

Probability ke axioms se shuru karte hain:

Axiom 1: Kisi bhi event $E$ ke liye, $0 \leq P (E) \leq 1$
Axiom 2: $P (S) = 1$ (kuch na kuch to hona chahiye)
Axiom 3: Agar events mutually exclusive hain, $P (A \cup B) = P (A) + P (B)$

Equally likely outcomes ke liye:

Agar sample space mein $n$ outcomes hain, sab equally likely, aur kuch na kuch hone ki probability 1 hai:

$P (outcome_{1}) + P (outcome_{2}) + \dots + P (outcome_{n}) = 1$

Kyunki sab equal hain, maano $P (each outcome) = p$ :

$n \cdot p = 1 ⟹ p = \frac{1}{n}$

Har outcome ki probability $\frac{1}{n ( S )}$ hai.

Ab agar event $E$ mein $k$ outcomes hain, Axiom 3 se (woh mutually exclusive hain):

$P (E) = k times \frac{1}{n ( S )} + \frac{1}{n ( S )} + \dots + \frac{1}{n ( S )} = \frac{k}{n ( S )} = \frac{n ( E )}{n ( S )}$

Isliye counting probability ke liye kaam karta hai!

4. Key Properties

5. Worked Examples with Step-by-Step WHY

6. Common Mistakes & Unhe Fix Karne Ka Tarika

7. Active Recall Practice

Recall Feynman Technique — Ek 12-Saal-Ke Bacche Ko Explain Karo

Imagine karo tum apne chhote sibling ko probability explain kar rahe ho:

"Theek hai, to probability guessing games jaisi hai. Maano tumhare paas ek box hai 10 toys ke saath—7 cars aur 3 dolls. Tum aankhein band karke ek pakad lete ho. Car milne ke chances kya hain?

Pehle, SAARI toys count karo: 10 total. Yeh sab kuch hai jo ho sakta tha.

Phir, jo tum CHAHTE ho woh count karo: 7 cars.

Tumhare chances = fraction: 10 mein se 7, ya 7/10. Woh 0.7 hai, ya 70%. Agar maine kaha 'Car nahi milne ka chance kya hai?' — to woh doll milna hai. 10 toys mein se 3 dolls = 3/10 ya 30%.

Dhyaan do: 70% + 30% = 100%. Kyunki tum ZAROOR kuch pao ge!

Probability bas hai: (jo tum chahte ho) ÷ (saari possibilities). Jab tak har toy ko equally pick hone ka chance hai, yeh perfectly kaam karta hai."

Connections

Sample Space — saare possible outcomes
Events and Set Theory — events as subsets, unions, intersections
Complement of an Event — jo nahi hota
Mutually Exclusive Events — saath nahi ho sakte
Axioms of Probability — foundation rules
Conditional Probability — extra information dene par probability
Counting Principles — complex sample spaces ke liye permutations aur combinations
Random Variables — numerical outcomes aur unki probabilities
Law of Large Numbers — probability → frequency with many trials

#flashcards/maths

Sample space kya hota hai? :: Kisi random experiment ke saare possible outcomes ka set.

Probability mein event kya hota hai?

Sample space ka ek subset; ek ya zyada outcomes ka collection jisme hum interested hain.

Classical probability formula kya hai?

P (E) = \frac{n ( E )}{n ( S )}

jahan

n (E)

favourable outcomes ki number hai aur

n (S)

total outcomes ki number hai, assuming saare outcomes equally likely hain.

Kisi bhi probability value ki range kya hoti hai?

0 \leq P (E) \leq 1

. Probability negative ya 1 se badi nahi ho sakti.

Complement rule kya hai?

P (E^{'}) = 1 - P (E)

jahan

E^{'}

woh event hai ki

E

occur nahi hota.

Agar fair die roll karo, to P(2 ya kam number aana) kya hai?

\frac{2}{6} = \frac{1}{3}

kyunki favourable outcomes {1, 2} hain 6 total outcomes mein se.

Do coins toss karne par sample space kya hoga?

{HH, HT, TH, TT} — chaar equally likely outcomes.

Do fair coins toss karne par P(exactly one head) kya hai?

\frac{2}{4} = \frac{1}{2}

kyunki {HT, TH} favourable outcomes hain 4 total mein se.

Do dice roll karne par sample space mein kitne outcomes hote hain?

36 outcomes (6 × 6), ordered pairs (1,1) se (6,6) tak.

Do dice roll karne par P(sum = 7) kya hai?

\frac{6}{36} = \frac{1}{6}

kyunki sum 7 paane ke 6 tarike hain: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}.

Hum hamesha outcomes count karke divide kyu nahi kar sakte?

Formula

P (E) = \frac{n ( E )}{n ( S )}

tabhi kaam karta hai jab sample space ke saare outcomes equally likely hon.

52 cards ki deck mein, P(heart draw karna) kya hai?

\frac{13}{52} = \frac{1}{4}

kyunki 52 cards mein 13 hearts hain.