The probability of guessing the correct answer of a certain test question is (x/12). If the probability of not guessing the correct answer is ( ⅚), then find the value of x.

 The probability of guessing the correct answer of a certain test question is (x/12). If the probability of not guessing the correct answer is ( ⅚), then find the value of x.

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At first glance, it may appear simple, but this problem is an excellent example of a question that checks conceptual understanding, numerical accuracy, and knowledge of complementary events in probability. It can appear in numerous exams worldwide, including CBSE, ICSE, IGCSE, GCSE, IB, AP, SAT, ACT, GRE, GMAT, and SOA actuarial examinations.

Step-by-Step Solution

To solve the problem, we start by recalling one of the fundamental rules in probability:

The sum of the probability of an event and the probability of its complement is always equal to one.

Let’s denote:

• $P(\text{correct}) = x/12$
  

• $P(\text{not correct}) = 5/6$
  

According to the complementary rule:

$$P(\text{correct}) + P(\text{not correct}) = 1$$

Substituting the given values:

$$x/12 + 5/6 = 1$$

To solve for $x$ first express 5/6 as a fraction with denominator 12:

$$5/6 = 10/12$$

So the equation becomes:

$$x/12 + 10/12 = 1$$

Combine like terms:

$$(x + 10)/12 = 1$$

Multiply both sides by 12 to eliminate the denominator:

$$x + 10 = 12$$

Subtract 10 from both sides:

$$x=2$$

Therefore, the probability of guessing the correct answer is 2/12, which simplifies to 1/6. This satisfies the given probability of not guessing the correct answer (5/6), since $1/6 + 5/6 = 1$.

Understanding Complementary Probability

The problem provides a perfect example of the complementary rule. In probability theory, every event has a complement — the set of outcomes where the event does not occur. The sum of the probabilities of an event and its complement always equals one.

This principle is fundamental in mathematics curricula worldwide. It is introduced in CBSE Class 9 as part of Chapter 15: Probability, where students learn about experimental probability using dice, coins, and simple card experiments. ICSE Class 9 and 10 also emphasize complement probability in their algebraic and practical problem sections. In IGCSE and GCSE mathematics, complement rules are a standard part of both foundation and higher-tier probability questions. IB Diploma students encounter similar questions in Analysis and Approaches or Applications and Interpretation courses, where both theoretical and experimental probability are explored.

Even in professional contexts such as AP Statistics, GRE, and GMAT quantitative sections, the complementary rule forms the basis for more complex probability problems, including conditional probability, joint probability, and expected value calculations. SOA actuarial exams often use complement probability as a foundational step before progressing to advanced stochastic models and actuarial risk assessments.