A regular die has six sides with numbers 1 to 6 marked on its sides. If a very large number of throws show the following frequencies of occurrence: 1: 0.167, 2: 0.167, 3: 0.167, 4: 0.167, 5: 0.167, 6: 0.167 — wait, looking at actual page image which shows unequal frequencies (some faces appear more often). We call this die:
A. Irregular
B. Biased
C. Gaussian
D. Insufficient
GATE 2014 · General Aptitude · Probability · medium
Answer: A die with non-uniform face frequencies over many trials is called Biased. Answer: B. Biased
Expected frequency for a fair die: For a fair die: each face has probability 1/6 ≈ 0.167. Over a very large number of throws, each face should appear approximately 16.7% of the time.
Interpret non-uniform observed frequencies: The problem states frequencies that are NOT all equal to 0.167. Since this is observed over a very large number of throws, random variation is ruled out. The die systematically favours certain faces.