Probability, Probability Graphs & Compound Bar Chartsप्रायिकता (सम्भावना)
Module 6 covers theoretical and empirical probability, probability graphs, and compound bar charts, carrying significant weight in the Super TET Junior Mathematics section. Mastering ratio formulations, mutually exclusive events, and comparative data interpretation from compound bar charts is essential for securing top ranks.
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Syllabus Focus & High-Yield Strategy: Module 6 covers theoretical and empirical probability, probability graphs, and compound bar charts, carrying significant weight in the Super TET Junior Mathematics section. Mastering ratio formulations, mutually exclusive events, and comparative data interpretation from compound bar charts is essential for securing top ranks.
1. Fundamental Axioms, Definitions & Foundational Properties
PEACE — Probability Equals Actual Count Exhaustive.2. Formulas, Standard Theorems & Algorithmic Procedures
| Category / Case | Formula / Mathematical Rule | Topper Calculation Shortcut |
|---|---|---|
| Classical Probability | $P(E) = rac{n(E)}{n(mathcal{S})}$ | Cancel common factors in numerator and denominator before final division. |
| Addition Rule (General) | $P(A cup B) = P(A) + P(B) - P(A cap B)$ | If $A$ and $B$ are mutually exclusive, $P(A cap B) = 0$, simplifying to $P(A) + P(B)$. |
3. Word Problem Modeling & Practical Applications
4. Mathematical Edge Cases, Sign Conventions & Common Traps
LEAST — Learn Exactly All Subtracted Total ($1 - P( ext{none})$).Comprehensive Mathematical Matrix & Method Comparison
| Parameter / Dimension | Method / Concept A | Method / Concept B | Exam Speed & Accuracy Takeaway |
|---|---|---|---|
| Event Relationship | Mutually Exclusive ($A cap B = emptyset$) | Independent Events ($P(A cap B) = P(A)P(B)$) | Do not confuse mutual exclusivity with independence; exclusive events with $P > 0$ are never independent. |
| Sampling Method | With Replacement | Without Replacement | Without replacement alters the denominator $n(mathcal{S})$ at every successive draw. |
| Bar Chart Type | Simple Bar Chart | Compound / Grouped Bar Chart | Simple charts show single metrics; compound charts compare sub-categories across shared baselines. |
| Calculation Approach | Classical Empirical Ratio | Complementary Subtraction Rule ($1 - P$) | Use $1 - P( ext{none})$ for 'at least one' problems to bypass tedious union calculations. |
1. What is the probability of getting a sum of 8 when two unbiased six-faced dice are rolled simultaneously?
Total sample space $n(\mathcal{S}) = 6 \times 6 = 36$. Favorable outcomes for a sum of 8 are $(2,6), (3,5), (4,4), (5,3), (6,2)$, giving $n(E) = 5$. Thus, $P(E) = \frac{5}{36}$. Option 1 is correct. Distractors like $\frac{1}{6}$ arise from miscounting favorable pairs.
2. From a well-shuffled pack of 52 playing cards, one card is drawn at random. What is the probability that it is either a king or a red card?
Let $K$ be the event of drawing a king ($n(K) = 4$) and $R$ be the event of drawing a red card ($n(R) = 26$). The intersection $K \cap R$ contains 2 red kings. Using the addition rule: $P(K \cup R) = P(K) + P(R) - P(K \cap R) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}$. Option 0 is correct.
3. A bag contains 5 red, 4 green, and 3 yellow marbles. If two marbles are drawn at random without replacement, what is the probability that both are red?
Total marbles = $5 + 4 + 3 = 12$. Probability of first red = $\frac{5}{12}$. After drawing one red, remaining red = 4, total marbles = 11. Probability of second red given first is red = $\frac{4}{11}$. Total probability = $\frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$. Option 0 is correct.
4. If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cup B) = 0.7$, what is the value of $P(A|B)$?
By inclusion-exclusion, P(A and B)=0.4+0.5-0.7=0.2. Thus P(A given B)=P(A and B)/P(B)=0.2/0.5=0.4=2/5. The answer is Option B.
5. Three unbiased coins are tossed simultaneously. What is the probability of getting at least two heads?
Sample space for 3 coins $n(\mathcal{S}) = 2^3 = 8$. Outcomes with at least two heads are {HHT, HTH, THH, HHH}, which equals 4 outcomes. Thus, $P = \frac{4}{8} = \frac{1}{2}$. Option 0 is correct.
6. In a compound bar chart comparing export data of two items (A and B) across years 2021 and 2022, if Item A's bar height goes from 50 to 75, what is the percentage increase?
Percentage increase = $\frac{75 - 50}{50} \times 100\% = \frac{25}{50} \times 100\% = 0.5 \times 100\% = 50\%$. Option 2 is correct. Distractor 25% is the absolute difference, and 33.33% uses 75 as the base incorrectly.
7. A card is drawn from a standard deck of 52 cards. What is the probability that it is neither a heart nor a face card?
There are 13 hearts and 12 face cards, with 3 cards counted in both groups. So 22 cards are hearts or face cards, leaving 52-22=30 cards that are neither. The probability is 30/52=15/26. The answer is Option A.
8. If the probability of a student passing the Super TET Mathematics exam is $0.78$, what is the probability that the student will not pass?
Probability of complementary event $P(\bar{E}) = 1 - P(E) = 1 - 0.78 = 0.22$. Option 1 is correct. Distractor 0.28 is a common subtraction error ($1.00 - 0.72$).
9. Two dice are thrown. What is the probability that the product of the numbers appearing on top is a prime number?
A product of two die results is prime only when one result is 1 and the other is 2, 3, or 5. There are 6 ordered outcomes out of 36, so the probability is 6/36=1/6. The answer is Option B.
10. In a compound bar chart representing school enrollment across years 2020 and 2021 for Classes VI, VII, and VIII, if total enrollment in 2020 was 450 and in 2021 was 540, what is the absolute difference represented as a ratio of 2020 enrollment?
Absolute difference = $540 - 450 = 90$. Ratio of absolute difference to 2020 enrollment = $\frac{90}{450} = \frac{1}{5}$. Option 0 is correct.
5-Minute Exam Simulation Mode
Experience the real pressure of Super TET Junior (Part 2C Mathematics). You will face 10 randomized speed-math questions with negative marking (-1 for incorrect answers) and a strict 5-minute countdown.
High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)
Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)
Probability Fundamentals Acronym
PEACE
P - Probability, E - Equals, A - Actual, C - Count, E - Exhaustive.
At Least One Rule Acronym
LEAST
L - Learn, E - Exactly, A - All, S - Subtracted, T - Total ($1 - P(\text{none})$).
Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)
Negative Marking Trap 1: Forgetting to subtract the intersection $P(A \cap B)$ in general union probability calculations.
Negative Marking Trap 2: Assuming draws 'without replacement' maintain constant denominators in successive fractions.
Negative Marking Trap 3: Misinterpreting compound bar chart legends and reading values from the wrong data series bar.
Negative Marking Trap 4: Confusing odds in favor ($a:b$) with probability $\frac{a}{a+b}$.