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compulsory · CSS study notes

3. Probability & Counting

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 3 (Probability)

Past Paper Weightage

  • Probability questions (dice/cards/balls) recur almost every year; counting (permutation/combination) is asked less often but as a moderate-scoring shortcut question.

Key Formula

P(E)=Number of favourable outcomesTotal number of possible outcomesP(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}
  • P(E) always lies between 0 and 1 (0 = impossible, 1 = certain)
P(not E)=1−P(E)P(\text{not } E) = 1 - P(E)

Counting Principles

  • Fundamental Counting Principle: if one event can occur in m ways and a second in n ways, both together can occur in m×n ways
  • Factorial:
n!=n×(n−1)×⋯×1n! = n\times(n-1)\times \dots \times 1

(counts the number of ways to arrange n distinct items in a row); 0! = 1

  • Permutation (order matters — arrangements):
nPr=n!(n−r)!{}^nP_r = \frac{n!}{(n-r)!}
  • Combination (order doesn't matter — selections):
nCr=n!r!(n−r)!{}^nC_r = \frac{n!}{r!(n-r)!}
  • Rule of thumb: "arrange/rank/order" → permutation; "choose/select/form a group" → combination

Quiz (with answers)

  1. Prime number on a single die roll (primes: 2, 3, 5) = 3/6 = ½ (B)
  2. A Queen from a standard 52-card deck (4 Queens) = 4/52 = 1/13 (B)
  3. An even number on a die roll (2, 4, 6) = 3/6 = ½ (B)

Worked Questions (with answers)

  1. 6 pillows (3 red, 2 yellow, 1 blue) — P(yellow) = 2/6 = 1/3
  2. P(rolling a 3 on a die) = 1/6
  3. Card drawn from 12 cards numbered 1–12:
    • (i) P("8") = 1/12
    • (ii) P(even number: 2,4,6,8,10,12) = 6/12 = 1/2
    • (iii) P(perfect square: 1, 4, 9) = 3/12 = 1/4
    • (iv) P(negative number) = 0 (impossible)
    • (v) P(number less than 13) = 12/12 = 1 (certain)

Practice Questions (with answers)

  1. Blood groups among 1,000 people: A=300, B=325, O=250, AB=125 → P(AB) = 125/1000 = 1/8 (0.125)
  2. Village blood groups: A=600, B=650, AB=550, O=200 (total 2,000):
    • (i) P(B) = 650/2000 = 13/40 (0.325)
    • (ii) P(O) = 200/2000 = 1/10 (0.1)
  3. Two-digit number (10–99) chosen at random; multiples of 7 in range = {14,21,...,98} = 13 numbers → P = 13/90
  4. Bag: 47 white + 5 yellow + 10 black = 62 balloons → P(black) = 10/62 ≈ 16.13%
  5. P(Head) with a fair coin = 1/2

Random Sampling

  • Population: the whole group being studied; sample: the part chosen from it; sampling frame: the list of all members
  • Random sampling: every member has a known, equal chance of being chosen, so the sample is unbiased
  • How to select randomly: a lottery or draw, a random number table, or a computer random-number generator
  • Probability of selecting a given member = n ÷ N (sample size ÷ population size)
MethodHow it works
Simple randomPure chance; with or without replacement
SystematicRandom start, then every k-th member (k = N ÷ n)
StratifiedSplit the population into groups (strata) and sample from each in proportion
ClusterRandomly choose whole groups (e.g. schools) and study all inside
  • Non-random (non-probability) sampling: convenience, judgment, quota, snowball; cheaper but may be biased
  • Sampling error vs bias: a larger random sample lowers error, but only a good method removes bias
  • Example: choose 10 students from 200 by systematic sampling: k = 200 ÷ 10 = 20; with a random start of 7 the sample is 7, 27, 47, …; each student's chance = 10 ÷ 200 = 5%
  • Past-paper link: sampling questions (CSS 2020 Q8(a), 2022 Q8(a)) are answered below

MCQ Traps

  • Probability of an impossible event is 0; of a certain event is 1 — don't leave these blank or guess a fraction
  • "With replacement" vs "without replacement" changes the total outcomes on subsequent draws — check the question wording carefully
  • Percentage-style probability answers should still be sanity-checked against the 0–1 (or 0–100%) range

Self-Test Questions

  1. Define probability and state its formula.
  2. What is the probability of rolling a number greater than 4 on a fair six-sided die?
  3. A bag has 4 red and 6 blue balls. Find the probability of drawing a red ball.
  4. Two coins are tossed. What is the probability of getting exactly one head?

Past-Paper MCQs (FPSC)

✅

Taken from the FPSC papers you shared (2015–2025). FPSC does not publish MCQ keys, so the answer is worked out by me.

  1. (CSS 2024, Q1.3) What are total number of combinations of 09 things taken 03 at a time? (A) 18 (B) 27 (C) 48 (D) 84
  • ✅ Answer key
    1. (D) 84, since ⁹C₃ = (9 × 8 × 7) ÷ (3 × 2 × 1) = 504 ÷ 6 = 84

Past-Paper Questions with Answers (FPSC)

✅

Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me.

  • CSS 2023 Q8(b): A card is drawn from 12 cards numbered 1–12. Find the probability of (i) '8', (ii) an even number, (iii) a perfect square, (iv) a negative number, (v) a number less than 13. (5)
    1. Total outcomes = 12
    2. (i) P(8) = 1/12
    3. (ii) Even numbers 2, 4, 6, 8, 10, 12 → 6/12 = 1/2
    4. (iii) Perfect squares 1, 4, 9 → 3/12 = 1/4
    5. (iv) P(negative) = 0 (impossible)
    6. (v) P(less than 13) = 12/12 = 1 (certain)
  • CSS 2020 Q8(b): In a village 600 people have blood group A, 650 B, 550 AB and 200 O. Find the probability of (i) group B, (ii) group O. (5)
    1. Total = 600 + 650 + 550 + 200 = 2,000
    2. (i) P(B) = 650/2000 = 13/40 = 0.325 (32.5%)
    3. (ii) P(O) = 200/2000 = 1/10 = 0.1 (10%)
  • CSS 2016 Q12(a): Of 1,000 people, 300 have blood group A, 325 B, 250 O and 125 AB. Probability that a random person has group AB? (5)
    1. P(AB) = 125/1000 = 1/8 = 0.125
  • CSS 2022 Q8(a), 2020 Q8(a): What is sampling? Discuss non-probability sampling; what is systematic sampling and its types? (5)
    • Sampling: choosing a part (sample) of a population to study and make conclusions about the whole
    • Probability sampling (every member has a known chance): simple random, systematic, stratified, cluster
    • Non-probability sampling (selection not random; chances unknown)
      • Convenience: whoever is easy to reach
      • Judgment/purposive: researcher picks typical cases
      • Quota: fixed numbers from each group
      • Snowball: participants recruit others
      • Weakness: cannot measure sampling error or generalize safely
    • Systematic sampling: pick a random start, then every k-th item, where k = N ÷ n
      • Types: linear systematic, circular systematic (wraps around the list); also variations such as repeated and multiple starts
      • Advantage: simple and spreads evenly; risk: hidden pattern in the list

CSS Exam Perspective

🎯

  • Most repeated areas: simple probability from tables (blood groups, cards), combinations, sampling types
  • Tip: P = favourable ÷ total; a probability is always between 0 and 1; write the total first