Mutually exclusive events — addition rule: Foundations

2.7.6 · D1 · Maths › Statistics & Probability — Intermediate › Mutually exclusive events — addition rule

Parent note ki ek bhi line pe trust karne se pehle, tumhe usmein har squiggle bina jhijhake padhni aani chahiye. Yeh page har ek cheez ko zero se banata hai, ek aisi order mein jahan har idea usse pehle wale pe lean karta hai.

1 — Ek "outcome": sab kuch ka atom

Die ek baar roll karo. Woh ek face dikhata hai. Woh face outcome hai. Tum ise aur split nahi kar sakte; yeh atom hai.

Yeh topic iski zaroorat kyun rakhta hai: har probability jo hum compute karte hain woh actually "us group mein kitne dots hain jo mujhe chahiye" hai. Agar tum outcomes ko countable dots ki tarah nahi dekhte, toh addition rule sirf symbols hai.

2 — Sample space $S$ aur count $n (\cdot)$

Figure — Mutually exclusive events — addition rule

Figure 1. Teal rectangle ek die roll ke liye poora sample space $S$ hai; har orange dot ek single outcome hai (faces 1–6). Plum arrow ek dot ki taraf point karta hai taaki yaad rahe "ek outcome = ek dot". Corner mein label $n (S) = 6$ ek counter hai jo chhe dots total report karta hai — yahi woh denominator hai jisse tum har probability ke liye divide karoge.

Symbol $n (\cdot)$ ek machine hai jo count karta hai ki tumne brackets mein jo daala hai usme kitne dots hain:

$n (S)$ = total kitne dots (die ke liye, $n (S) = 6$ ).
$n (A)$ = group $A$ mein kitne dots hain.

Yeh topic iski zaroorat kyun rakhta hai: parent ka sabse pehla line $P (E) = \frac{n ( E )}{n ( S )}$ hai. Woh poora formula bas "chahiye wale dots over total dots" hai. $n$ counter hai.

3 — Ek "event": dots ka ek bounded group

Subset ka word literally matlab hai " $S$ ke andar se liya gaya ek group". Event mein har dot $S$ mein bhi hai; ek event kabhi naye outcomes nahi banata.

Curly braces ${}$ ka matlab bas "in listed dots ko contain karne wala collection" hai.

Yeh topic iski zaroorat kyun rakhta hai: addition rule do events $A$ aur $B$ ko combine karta hai. Sab kuch dots ke around ek fence hai.

4 — Plus $+$ aur minus $-$ : combine karo aur hatao

Kisi bhi formula se pehle, poore note ke do sabse simple symbols ko samjho.

Yeh topic iski zaroorat kyun rakhta hai: aage ka har formula ( $P (A) + P (B)$ , $1 - P (A)$ , $P (A) + P (B) - P (A \cap B)$ ) sirf inhi do moves se bana hai. Agar $-$ ek mystery hai, toh general rule ek mystery hai.

5 — Probability $P (E)$ : fair counting

Letter $P$ ko "the probability of" padha jaata hai. Yeh hamesha $0$ aur $1$ ke beech ek number deta hai:

$P = 0$ matlab event mein koi dot nahi — yeh kabhi nahi ho sakta.
$P = 1$ matlab event mein saare dots hain — yeh hamesha hota hai.
$P = \frac{1}{2}$ matlab aadhe dots tumhare hain.

Yeh topic iski zaroorat kyun rakhta hai: poora addition rule pehle dots count karke, phir end mein sab kuch $n (S)$ se divide karke derive hota hai. $P$ bas counter $n$ hai jisme total divide ho chuka hai.

6 — Union $\cup$ : "OR" word ek picture ke roop mein

Figure 2. Do overlapping fences: orange disc event $A$ hai, teal disc event $B$ hai. Jo bhi shaded hai — orange part, teal part, aur beech ka lens jis par "both" likha hai — woh union $A \cup B$ hai. Beech mein lens dekho: woh dots dono fences mein belong karte hain, yahi exact reason hai kyun blindly "orange dots + teal dots" add karne se unhe twice count kiya jaata. Yeh picture woh reason hai kyun general rule mein subtraction chahiye.

Symbol $\cup$ ek cup ki tarah dikhta hai jo dono fences se sab kuch scoop karke ek bade fence mein le jaata hai. $\cup$ ko zyoor se "or" padho.

Yeh topic iski zaroorat kyun rakhta hai: addition rule exactly ek sawaal ka jawaab deta hai: $P (A \cup B)$ kya hai? — chance ki do events mein se kam se kam ek hogi.

7 — Intersection $\cap$ aur empty set $\emptyset$

Symbol $\cap$ ek ulta cup hai — ek cap — aur $\cap$ ko "and" padha jaata hai. Yeh sirf shared dots rakhta hai.

Yeh topic iski zaroorat kyun rakhta hai: "mutually exclusive" define hota hai $A \cap B = \emptyset$ se. Woh ek line kehti hai "do fences koi dot share nahi karti", toh unka overlap empty hai, toh double-count karne ke liye kuch nahi.

8 — Sab ko saath rakhna: general addition rule

Ab jab $+$ , $-$ , $\cup$ aur $\cap$ sab ka matlab samajh aa gaya, hum woh master formula state kar sakte hain jis par poora parent note depend karta hai.

Yeh topic iski zaroorat kyun rakhta hai: yeh woh rule hai jisse parent special case "earn" karta hai. Mutually-exclusive shortcut bas yahi formula hai jismein overlap zero ke barabar hai.

9 — Mutually exclusive: fences jo kabhi nahi milti

Figure 3. Yahan orange fence $A$ aur teal fence $B$ ek clear gap ke saath alag baitha hai (plum arrow) — koi dot shared nahi, toh $A \cap B = \emptyset$ . Figure 2 se compare karo, jisme ek "both" lens tha; woh lens gayab ho gaya. Koi overlap hatane ke liye nahi hai, toh general rule mein subtraction term zero hai aur plain addition bilkul sahi hai.

Yeh topic iski zaroorat kyun rakhta hai: yeh poora topic ek condition mein hai. "Kya dono ek saath ho sakte hain?" — agar nahi, toh overlap $\emptyset$ hai aur tum bas add karo; agar haan, toh pehle overlap subtract karo.

Ek trial ka ek akela result — woh atom jo "light up" hota hai.

S

ka matlab kya hai aur

n (S)

kya count karta hai?

S

sample space hai (saare outcomes);

n (S)

count karta hai ki total mein kitne outcomes hain.

Dot-picture terms mein ek event kya hota hai?

S

ka ek subset — kuch outcomes ke around khinchi gayi ek fence jinka tum care karte ho.

+

aur

-

signs dot-counts ke saath kya karte hain?

+

counts ko saath rakhta hai;

-

ek count hatata hai (double-counted overlap hatane ke liye use hota hai).

Equally likely outcomes ke liye probability formula likho.

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

= chahiye wale dots over total dots.

Probability kabhi

1

se zyada kyun nahi ho sakti?

Tum jo exist karte hain unse zyada dots nahi chahh sakte, toh

n (E) \leq n (S)

fraction ko

\leq 1

force karta hai.

A \cup B

ka matlab kya hai aur

\cup

ko kaise padhte hain?

A

B

ya dono ke saare dots;

\cup

ko "or" padho (yeh collect karta hai).

A \cap B

ka matlab kya hai aur

\cap

ko kaise padhte hain?

Sirf

A

aur

B

dono mein wale dots;

\cap

ko "and" padho (overlap).

\emptyset

kya hai aur

P (\emptyset)

kya hai?

Empty set — zero dots wala group;

P (\emptyset) = 0

General addition rule state karo.

P (A \cup B) = P (A) + P (B) - P (A \cap B)

— dono add karo, overlap ek baar subtract karo.

Mutually exclusive ko symbols mein aur picture mein define karo.

A \cap B = \emptyset

— do fences koi dot share nahi karti, woh kabhi nahi milti.

Mutually exclusive events ke liye plain addition kyun kaam karta hai?

Overlap empty hai toh

P (A \cap B) = 0

; subtraction term gayab ho jaata hai,

P (A) + P (B)

bachta hai.

A^{'}

kya hai aur usse

A

se do kya properties link karti hain?

A

mein nahi wale saare dots;

A

aur

A^{'}

mutually exclusive aur exhaustive hain, toh

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

\sum_{i = 1}^{n} P (A_{i})

ka matlab kya hai?

P (A_{1}) + P (A_{2}) + \dots + P (A_{n})

add karo, ek term per counter value.

Connections

Mutually exclusive events — addition rule
Probability — basic definitions & sample space
Venn diagrams in probability
Complementary events
General addition rule & inclusion–exclusion

Foundations — Mutually exclusive events — addition rule

1 — Ek "outcome": sab kuch ka atom

2 — Sample space $S$ aur count $n (\cdot)$

3 — Ek "event": dots ka ek bounded group

4 — Plus $+$ aur minus $-$ : combine karo aur hatao

5 — Probability $P (E)$ : fair counting

6 — Union $\cup$ : "OR" word ek picture ke roop mein

7 — Intersection $\cap$ aur empty set $\emptyset$

8 — Sab ko saath rakhna: general addition rule

9 — Mutually exclusive: fences jo kabhi nahi milti

10 — Complement $A^{'}$ : fence ke bahar sab kuch

11 — Sigma $\sum$ : "bahut saare pieces ko add karo"

Prerequisite map

Equipment checklist

Connections