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Date Start: 01 Oct 2026, 08:00 AM
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Mathematical Risk vs Reward Optimization for 1/3 Negative Marking in CBT
1. Abstract & Statistical Foundation
Modern competitive CBT examinations (e.g., RRB NTPC, SSC CGL, BPSC) incorporate negative marking ratios, predominantly 1/3 (-0.33 mark penalty for an incorrect response vs +1.0 mark for a correct response). This empirical study models decision thresholds for question abandonment vs strategic elimination.
2. Expected Value Model ($E(X)$)
Let $n$ be the number of choices ($n = 4$).
Let $k$ be the number of options conclusively eliminated by the candidate ($0 \le k \le 3$).
The probability of selecting the correct choice uniformly at random among the remaining $4 - k$ options is:
$$P(\text{correct}) = \frac{1}{4 - k}$$
$$P(\text{incorrect}) = \frac{3 - k}{4 - k}$$
The Expected Value formula:
$$E(X) = 1.0 \times \frac{1}{4 - k} - 0.333 \times \frac{3 - k}{4 - k}$$
Evaluation of Expected Value Matrix:
$$E(X) = 1.0(0.25) - 0.333(0.75) = 0.250 - 0.250 = 0.00\text{ marks}$$
Decision: Zero expected advantage. Indiscriminate guessing introduces high variance without mathematical gain.
$$E(X) = 1.0(0.333) - 0.333(0.667) = 0.333 - 0.222 = +0.111\text{ marks}$$
Decision: Statistically positive expectation. Attempt recommended over large test samples.
$$E(X) = 1.0(0.500) - 0.333(0.500) = 0.500 - 0.167 = +0.333\text{ marks}$$
Decision: Strong statistical mandate to attempt. Across 30 such questions, expected net gain is $+10.0$ marks.