Probability – Quick Revision Notes (JKSSB Social Forestry Worker – Basic Mathematics)
1. What is Probability?
- Definition: Probability is a number between 0 and 1 that measures the likelihood (or chance) that a particular event will occur in a random experiment.
- Interpretation:
- 0 → impossible event
- 0.5 → equally likely to happen or not happen
- 1 → certain event
- Notation: Probability of an event A is written as P(A).
2. Basic Terminology
| Term | Meaning | Example |
|---|---|---|
| Experiment | Any process that yields an outcome. | Tossing a coin, drawing a card. |
| Outcome | A single possible result of an experiment. | Getting “Head” on a coin toss. |
| Sample Space (S) | Set of all possible outcomes. | S = {H, T} for a coin. |
| Event (E) | Any subset of the sample space (may contain one or more outcomes). | Getting at least one head in two tosses = {HH, HT, TH}. |
| Elementary Event | An event containing exactly one outcome. | {HH}. |
| Complementary Event (A’) | All outcomes not in A. | If A = “getting a head”, A’ = “not getting a head”. |
| Mutually Exclusive (Disjoint) | Two events cannot occur together (A ∩ B = ∅). | Getting a head and a tail on the same toss. |
| Exhaustive | A collection of events whose union equals the sample space. | {Head, Tail} are exhaustive for a coin toss. |
| Independent | Occurrence of one does not affect the probability of the other. | Tosses of a fair coin. |
| Dependent | Probability of one changes when the other is known. | Drawing cards without replacement. |
3. Approaches to Probability
| Approach | When to Use | Formula / Idea |
|---|---|---|
| Classical (Theoretical) | All outcomes equally likely. | P(A) = Number of favorable outcomes / Total number of outcomes |
| Empirical (Experimental / Relative Frequency) | Based on actual trials or data. | P(A) ≈ (Number of times A occurred) / (Total number of trials) |
| Subjective | Personal judgment when no data or symmetry exists. | No fixed formula; relies on expert opinion. |
Key Highlight: For JKSSB basic maths, the classical approach is most common (dice, cards, coins, simple selection problems).
4. Probability Axioms (Kolmogorov) – Quick Recall
- Non‑negativity: 0 ≤ P(A) ≤ 1 for any event A.
- Normalization: P(S) = 1 (probability of the whole sample space is 1).
- Additivity: If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).
Mnemonic: N.A.A. → Non‑negativity, Additivity, Axiom of All (sample space = 1).
5. Core Formulas
| Concept | Formula | When to Use |
|---|---|---|
| Probability of Complement | P(A′) = 1 – P(A) | Easy to find “not A”. |
| Addition Rule (General) | P(A ∪ B) = P(A) + P(B) – P(A ∩ B) | Any two events (overlap subtracted). |
| Addition Rule (Mutually Exclusive) | P(A ∪ B) = P(A) + P(B) | When A and B cannot happen together. |
| Multiplication Rule (General) | P(A ∩ B) = P(A) × P(B│A) | Probability of both A and B. |
| Multiplication Rule (Independent) | P(A ∩ B) = P(A) × P(B) | When A and B do not influence each other. |
| Conditional Probability | P(A│B) = P(A ∩ B) / P(B) , P(B) > 0 | “Probability of A given B has occurred”. |
| Odds in Favor | Odds = P(A) : P(A′) | Useful in betting‑type questions. |
| Odds Against | Odds against = P(A′) : P(A) | Reverse of odds in favor. |
6. Mnemonics for Quick Recall
- “AND → Multiply, OR → Add (if mutually exclusive)” – AND = intersection → use multiplication rule.
- OR = union → use addition rule; subtract overlap if not mutually exclusive.
- “Complement = 1 – Probability” – think of C as “take away from 1”.
- “Independent → ‘No Influence’ → Multiply plainly” – if events do not affect each other, just multiply their probabilities.
- “Conditional → ‘Given’ → Divide by the given event’s probability” – P(A│B) = P(A∩B) / P(B).
- “Exhaustive = Whole Pie” – if events are exhaustive, their probabilities sum to 1 (like slices of a whole pizza).
- “Mutually Exclusive = No Overlap” – picture two non‑overlapping circles; addition is simple.
7. Worked‑Out Examples (Typical JKSSB Style)
Example 1 – Simple Classical Probability
Question: A bag contains 5 red, 3 green, and 2 blue balls. One ball is drawn at random. Find the probability that the ball is not green.
Solution:
- Total outcomes = 5 + 3 + 2 = 10.
- Favourable outcomes for “not green” = red + blue = 5 + 2 = 7.
- P(not green) = 7/10 = 0.7.
Alternate using complement:
P(green) = 3/10 → P(not green) = 1 – 3/10 = 7/10.
Example 2 – Addition Rule (Non‑mutually Exclusive)
Question: In a class of 40 students, 25 like Mathematics, 20 like Science, and 12 like both. Find the probability that a randomly selected student likes Mathematics or Science.
Solution:
- P(M) = 25/40, P(S) = 20/40, P(M∩S) = 12/40.
- P(M ∪ S) = P(M) + P(S) – P(M∩S) = (25+20‑12)/40 = 33/40 = 0.825.
Example 3 – Multiplication Rule (Independent)
Question: A fair die is rolled twice. What is the probability of getting a 4 on the first roll and an odd number on the second roll?
Solution:
- P(4 on first) = 1/6.
- P(odd on second) = 3/6 = 1/2 (since 1,3,5 are odd).
- Independent → P = (1/6) × (1/2) = 1/12 ≈ 0.0833. #### Example 4 – Conditional Probability
Question: From a deck of 52 cards, two cards are drawn without replacement. Find the probability that the second card is a king given that the first card was a king.
Solution:
- After first king removed, 51 cards remain, 3 of them are kings.
- P(King₂ │ King₁) = 3/51 = 1/17 ≈ 0.0588. #### Example 5 – Bayes’ Theorem (Optional for JKSSB)
Question: Box A contains 3 red & 2 blue balls; Box B contains 1 red & 4 blue balls. A box is chosen at random and a ball is drawn; it turns out to be red. What is the probability that the ball came from Box A?
Solution:
- P(A) = P(B) = 1/2.
- P(Red│A) = 3/5, P(Red│B) = 1/5.
- P(Red) = P(A)P(Red│A) + P(B)P(Red│B) = (1/2)(3/5)+(1/2)(1/5)= (3+1)/10 = 4/10 = 2/5.
- By Bayes: P(A│Red) = [P(A)P(Red│A)] / P(Red) = [(1/2)(3/5)] / (2/5) = (3/10)/(2/5)= (3/10)*(5/2)=15/20=3/4=0.75. —
8. Common Pitfalls & How to Avoid Them
| Mistake | Why it Happens | Correct Approach |
|---|---|---|
| Adding probabilities for non‑mutually exclusive events without subtracting overlap | Forgetting the intersection term. | Always use P(A ∪ B) = P(A) + P(B) – P(A ∩ B). |
| Multiplying probabilities for dependent events as if they were independent | Assuming independence without checking. | Use P(A ∩ B) = P(A) × P(B│A) or compute directly from reduced sample space. |
| Confusing “odds” with “probability” | Odds are ratios, not fractions. | Remember: Odds in favor = P(A) : (1‑P(A)). Convert if needed. |
| Misplacing the condition in conditional probability | Writing P(A│B) as P(B│A). | Keep the given event after the vertical bar; it’s the denominator. |
| Using the wrong total number of outcomes | Miscounting sample space (especially with replacement vs. without). | List or compute sample space carefully; for replacement, multiply possibilities; for without, reduce counts after each draw. |
| Neglecting complement when it simplifies the problem | Doing lengthy counting when “not” is easier. | If asked for “at least one”, often compute 1 – P(none). |
9. Quick Reference Tables #### 9.1. Probability of Common Experiments
| Experiment | Total Outcomes | Favourable (Example) | Probability |
|---|---|---|---|
| Toss a fair coin | 2 | Head | 1/2 |
| Roll a fair die | 6 | Even number (2,4,6) | 3/6 = 1/2 |
| Draw a card from 52‑card deck | 52 | Ace | 4/52 = 1/13 |
| Draw 2 cards without replacement | 52×51 = 2652 | Both Aces | (4/52)*(3/51)=1/221 |
| Roll two dice | 36 | Sum = 7 | 6/36 = 1/6 |
| Spin a spinner with 4 equal sectors | 4 | Landing on sector 2 | 1/4 |
9.2. Frequently Used Probability Values
| Fraction | Decimal | Approx. % |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 2/3 | 0.666… | 66.7% |
| 3/4 | 0.75 | 75% |
| 1/6 | 0.166… | 16.7% |
| 5/6 | 0.833… | 83.3% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
9.3. Logical Flow for Solving a Probability Problem
- Identify the experiment and write down the sample space (S).
- Define the event(s) of interest (A, B, …).
- Determine the type of event(s): mutually exclusive? independent? exhaustive?
- Choose the appropriate approach: classical (counting), empirical (given data), or subjective. 5. Apply relevant formulas: complement, addition, multiplication, conditional, Bayes (if needed). 6. Simplify the fraction; convert to decimal or percentage if required.
- Check: Does the answer lie between 0 and 1? Does it make intuitive sense?
10. Key Highlights (Bullet Form)
- Probability range: 0 ≤ P(E) ≤ 1.
- Sum of probabilities of all elementary events = 1.
- Complement rule: P(E′) = 1 – P(E).
- Addition rule:
- General: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
- Mutually exclusive: P(A ∪ B) = P(A) + P(B).
- Multiplication rule:
- General: P(A ∩ B) = P(A) × P(B│A).
- Independent: P(A ∩ B) = P(A) × P(B).
- Conditional probability: P(A│B) = P(A ∩ B) / P(B) (provided P(B) > 0).
- Odds:
- In favor of A = P(A) : P(A′).
- Against A = P(A′) : P(A).
- Independent vs. Mutually Exclusive:
- Independent ⇒ P(A ∩ B) = P(A)P(B) (can happen together).
- Mutually exclusive ⇒ P(A ∩ B) = 0 (cannot happen together).
- Exhaustive events: If E₁, E₂,…,Eₙ are exhaustive, then P(E₁)+P(E₂)+…+P(Eₙ)=1.
- “At least one” problems: Often easier via complement: P(at least one) = 1 – P(none).
- “Exactly k” problems (binomial‑type): Use combinations when repetitions and independent trials are involved (though full binomial formula may be beyond basic scope, the idea of counting favorable outcomes remains).
- Replacement vs. Without Replacement: – With replacement → probabilities stay same each trial.
- Without replacement → update sample space after each draw (conditional probability).
11. Revision Checklist (Before the Exam)
- [ ] Can you define probability, experiment, outcome, sample space, event?
- [ ] Do you know the three axioms of probability?
- [ ] Are you comfortable with complement, addition, and multiplication rules?
- [ ] Can you distinguish between mutually exclusive and independent events?
- [ ] Do you know how to compute conditional probability and when to use Bayes’ theorem (if asked)? – [ ] Have you practiced problems involving coins, dice, cards, and simple draws from bags?
- [ ] Are you able to convert between probability, odds, and percentages? – [ ] Do you remember to check that your final answer lies between 0 and 1?
- [ ] Have you reviewed common pitfalls (over‑counting, forgetting to subtract intersection, mixing up independent vs. mutually exclusive)?
Final Tip: In the JKSSB Social Forestry Worker exam, the mathematics section is usually objective and time‑bound. Memorize the core formulas, practice a few quick‑calculation problems each day, and always verify that your answer is logical (e.g., probability of drawing a red card from a full deck cannot exceed 0.5). With these notes at your fingertips, you should be able to tackle any probability question with confidence. Good luck!
