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Noteclerical

Probability

Lesson 74 of 113

1. Ye topic hai kya?

Experiment = jo karo (sikka, pansa, pack se ek patta). Har possible result = outcome. Poori list = sample space S, n(S) kitne. Jo chahiye = event E. Class-X: outcomes equally likely (koi side bhari nahi).

Yahan lock

P(E) = n(E) / n(S). Simplify. 0 ≤ P(E) ≤ 1. Sure = 1, impossible = 0. P(E′) = 1 − P(E). “At least” = 1 − none.

Yahan kya nahi

Grouped mean / ogive. Dice opposite, net, n³ paint. Bayes / P(A|B) Class-XI. Two cards bina replacement (usually XI). Number-series agla term.

Pehle S likho (ya n(S) yaad: coin 2, 2 coins 4, 3 coins 8, die 6, 2 dice 36, pack 52). Phir favourable gino. Fraction chhota karo. 3/6 ko 1/2. Answer 0.5 bhi, paper aksar a/b.
Experimental: 80 toss mein 24 six → 24/80 = 0.3. Theoretical fair die 1/6. Lambe trial pe kareeb aate. Statistics frequency table se mean nahi — yahan count/total.

2. Sikka — S ko list

Ek coin: S = {H, T}, n=2. Do coins: 4 outcomes, HT aur TH alag (pehla/doosra). Teen coins: 8. “HH, ek-ek, TT” ko 3 equal mat maan-na — HH 1/4, mixed 1/2, TT 1/4.

HH HT TH TT n(S)=4 exactly one H

Highlight = exactly one head (2/4 = 1/2). At least one H = HH+HT+TH = 3/4 = 1 − P(TT).

2 coins: P(both H) = 1/4. P(same) = HH+TT = 1/2. P(different) = 1/2. 3 coins: P(exactly 2 H) = {HHT, HTH, THH} = 3/8. P(all H) = 1/8.
Example

Do fair coins. P(at least one head)?

Solution

  1. S

    HH, HT, TH, TT. n(S)=4. Favourable 3. 3/4.

  2. Complement

    1 − P(TT) = 1 − 1/4 = 3/4. Same. Trap: 2/3 (teen boxes). AP H,T,H,T agla nahi.

Answer3/4

3. Pansa — 6 aur 36

Ek fair die: S = {1,2,3,4,5,6}. Even {2,4,6}. Odd {1,3,5}. Prime on a die: 2, 3, 5 (1 prime nahi). 2 dice: pehla × doosra = 36 ordered pairs. (1,2) ≠ (2,1).

1 2 3 4 5 6

Even faces highlighted. P(even)=3/6=1/2. Opposite-sum-7 counting Dice chapter pe — yahan har face 1/6. P(6)=1/6. P(>4)={5,6}=1/3. P(multiple of 3)={3,6}=1/3.

2nd → 1st 1 2 3 4 5 6 1 1,6 2 2,5 3 3,4 4 4,3 5 5,2 6 6,1

Diagonal = sum 7, chhe cells, 6/36 = 1/6. Sum 2 ya 12: sirf 1/36. Double (same): 6/36 = 1/6. Ye 6×6 net fold nahi.

Example

Do dice. P(sum = 8)? P(sum ≥ 11)?

Solution

  1. Sum 8

    (2,6)(3,5)(4,4)(5,3)(6,2) — 5 ways. (1,7) nahi. 5/36. Sum 7 ke 6 se mix mat.

  2. ≥11

    Sum 11: (5,6)(6,5) = 2. Sum 12: (6,6)=1. Total 3/36 = 1/12.

Answer5/36 ; 1/12
Ordinary die pe opposite 1–6, 2–5, 3–4 — reasoning Dice/cube. Yahan har face 1/6, chahe opposite kya ho. “Painted 27 cubes” yahan nahi.

4. Pack aur bag

Standard pack: 52, 4 suits (♠♣♥♦), har suit 13 (A,2–10,J,Q,K). Red 26, black 26. Face cards Class-X: J, Q, K = 12 (Ace face nahi — 16 mat). Honour sometimes A+JQK = 16, paper “face” likhe to 12.

Eventn(E)P
Ace44/52 = 1/13
Heart / koi suit1313/52 = 1/4
King of hearts11/52
Red king21/26
Face (J,Q,K)1212/52 = 3/13
Black face66/52 = 3/26
Not ace4812/13
Example

Ek patta. P(diamond face card)? P(king or queen)?

Solution

  1. Diamond J,Q,K

    3/52. Poore face 12/52 nahi, poore diamond 13/52 nahi.

  2. K or Q

    4+4=8 (disjoint). 8/52 = 2/13. Ace mat jodna.

Answer3/52 ; 2/13
Example

Bag: 7 white, 5 black, 3 red. Ek ball. P(black)? P(not red)?

Solution

  1. n(S)

    7+5+3=15.

  2. Black

    5/15 = 1/3.

  3. Not red

    12/15 = 4/5, ya 1 − 3/15 = 4/5. Mean of 7,5,3 Statistics nahi.

Answer1/3 ; 4/5
Numbers 1 se 20 random: P(multiple of 3) = {3,6,9,12,15,18}=6/20=3/10. P(prime) gino 2,3,5,7,11,13,17,19 = 8/20=2/5. 1 prime nahi.

5. Complementary, sure/impossible, trial

E′ = “E nahi”. S ke har outcome ya E ya E′. Mutually exclusive + exhaustive: P(E)+P(E′)=1. Sure event: poora S, P=1 (die pe 1–6 mein se koi number). Impossible: empty, P=0 (die pe 7, pack se 2 of hearts jab “ek ace” nahi — die pe 7).

P(E′) = 1 − P(E). “At least one” = 1 − “zero”. “All not” wahi E′. P kabhi 7/6 ya −1 nahi. Do disjoint events: P(A or B)=P(A)+P(B) (king or queen). Overlap ho to Class-XI minus intersection — Class-X usually disjoint wording.
Example

Die. P(not 6)? 3 coins, P(at least one tail)?

Solution

  1. Not 6

    1 − 1/6 = 5/6. Direct 5/6.

  2. 3 coins

    n(S)=8. Zero tail = HHH only. At least one T = 1 − 1/8 = 7/8.

Answer5/6 ; 7/8
Example

Die 120 dafa, 18 baar 4 aaya. Experimental P(4)? Theoretical se farq?

Solution

  1. Exp

    18/120 = 3/20 = 0.15.

  2. Theory

    1/6 ≈ 0.167. Close, equal zaroori nahi. Ogive / median nahi.

Answer3/20 ; theory 1/6
“Random month, P(February)” = 1/12, 29 Feb Calendar pe leap-check. Yahan equally likely 12 months. Spinner 8 equal: P(ek sector)=1/8.

6. Sawal — basic se pro

Q1 · Basic

Fair coin. P(T)?

  1. 1/2
  2. 1
  3. 0
  4. 2

Solution

  1. 1/2

    D P>1 impossible.

Answer1/2
Q2 · Basic

Die. P(even)?

  1. 1/2
  2. 1/3
  3. 1/6
  4. 3

Solution

  1. 3/6

    1/2. B odd-prime mix. D count not probability.

Answer1/2
Q3 · Basic

Die. P(prime)?

  1. 1/2
  2. 2/3
  3. 1/6
  4. 4/6

Solution

  1. 2,3,5

    3/6=1/2. 1 aur 4 nahi. D agar 1 jod diya.

Answer1/2
Q4 · Basic

Do coins. P(exactly one H)?

  1. 1/2
  2. 1/4
  3. 3/4
  4. 1/3

Solution

  1. HT,TH

    2/4=1/2. B both H. C at least one. D teen-box trap.

Answer1/2
Q5 · Medium

Do dice. P(sum 7)?

  1. 1/6
  2. 1/36
  3. 7/36
  4. 5/36

Solution

  1. 6/36

    1/6. B sum 2. D sum 8. C number 7 ko numerator.

Answer1/6
Q6 · Medium

Pack 52. P(ace)?

  1. 1/13
  2. 1/52
  3. 4/13
  4. 1/4

Solution

  1. 4/52

    1/13. B one specific ace. D a suit.

Answer1/13
Q7 · Medium

Pack. P(face card J,Q,K)?

  1. 3/13
  2. 1/13
  3. 4/13
  4. 12/13

Solution

  1. 12/52

    3/13. C agar Ace joda (16/52). B sirf kings.

Answer3/13
Q8 · Medium

Bag 7W, 5B, 3R. P(not red)?

  1. 4/5
  2. 1/5
  3. 1/3
  4. 3/15

Solution

  1. 12/15

    4/5. B P(red). C P(black).

Answer4/5
Q9 · Pro

3 coins. P(at least one tail)?

  1. 7/8
  2. 1/8
  3. 1/2
  4. 3/8

Solution

  1. 1−HHH

    7/8. B all heads. D exactly two H.

Answer7/8
Q10 · Pro

Grouped table se median 155/6. Is chapter?

  1. Nahi — Statistics
  2. Haan, P(median)
  3. Die n/2
  4. Face cards

Solution

  1. cf formula

    A. Yahan n(E)/n(S).

AnswerA

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

Die. P(6)?

  1. 1/6
  2. 1/2
  3. 1
  4. 6/6

Solution

  1. 1/6

    D sure event.

Answer1/6
P2

Die. P(not 6)?

  1. 5/6
  2. 1/6
  3. 6/5
  4. 0

Solution

  1. 1−1/6

    5/6. C P>1.

Answer5/6
P3

Do coins. P(at least one H)?

  1. 3/4
  2. 1/2
  3. 1/4
  4. 1

Solution

  1. 1−TT

    3/4. B exactly one. C both H.

Answer3/4
P4

Do dice. P(sum 8)?

  1. 5/36
  2. 6/36
  3. 8/36
  4. 1/6

Solution

  1. 5 ways

    5/36. B sum 7.

Answer5/36
P5

Do dice. P(double)?

  1. 1/6
  2. 1/36
  3. 1/2
  4. 6

Solution

  1. (1,1)…(6,6)

    6/36=1/6. B sirf (6,6).

Answer1/6
P6

Pack. P(heart)?

  1. 1/4
  2. 1/13
  3. 13
  4. 1/52

Solution

  1. 13/52

    1/4.

Answer1/4
P7

Pack. P(king or queen)?

  1. 2/13
  2. 1/13
  3. 8
  4. 3/13

Solution

  1. 8/52

    2/13. D face teeno.

Answer2/13
P8

1 to 20. P(multiple of 3)?

  1. 3/10
  2. 1/3
  3. 3/20
  4. 6

Solution

  1. 6/20

    3/10. B 20/3 soch.

Answer3/10
P9

120 rolls, 18 fours. Experimental P(4)?

  1. 3/20
  2. 1/6
  3. 18
  4. 120/18

Solution

  1. 18/120

    3/20. B theoretical.

Answer3/20
P10

3×3×3 paint, exactly 2 faces. Is chapter?

  1. Nahi — Dice / cube
  2. Haan, P=12/27
  3. Two dice 36
  4. Mode

Solution

  1. Paint-count

    A. B galti se probability bana diya. Yahan fair die 1/6.

AnswerA

← Index · Statistics

Agle topic: Number system (Class-X arithmetic).