A man is known to speak truth 8 out of 11 times. He takes out two balls from a bag containing 3 white, 2 black and 5 red balls. What is the probability that he reports both the balls to be white?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{29}{165} \)
Note: The given answer is as per the official NTA Answer Key. However, the answer given by NTA appears to be incorrect, and the correct answer should be \( \displaystyle\frac{10}{33} \), which is not among the options, as explained below.
The bag has 3 + 2 + 5 = 10 balls. The chance that the two balls drawn are both white is
\( \displaystyle\frac{{}^{3}C_{2}}{{}^{10}C_{2}} = \displaystyle\frac{3}{45} = \displaystyle\frac{1}{15} \), so the chance that they are not both white is \( \displaystyle\frac{14}{15} \).
He reports “both white” in two ways:
They are both white and he speaks the truth: \( \displaystyle\frac{1}{15}\times\displaystyle\frac{8}{11} = \displaystyle\frac{8}{165} \)
They are not both white and he lies: \( \displaystyle\frac{14}{15}\times\displaystyle\frac{3}{11} = \displaystyle\frac{42}{165} \)
P(he reports both white) \( = \displaystyle\frac{8}{165}+\displaystyle\frac{42}{165} = \displaystyle\frac{50}{165} = \displaystyle\frac{10}{33} \)
Option (1) \( \displaystyle\frac{8}{21} \) — Incorrect. It does not come from any of the cases above.
Option (2) \( \displaystyle\frac{16}{29} \) — Incorrect. It does not come from any of the cases above.
Option (3) \( \displaystyle\frac{29}{165} \) — The answer given by NTA, but it seems to be incorrect. It counts a false report only when exactly one of the two balls is white (\( \frac{21}{45}\times\frac{3}{11} = \frac{21}{165} \)). It leaves out the draws with no white ball, where a lie also gives the report “both white”.
Option (4) \( \displaystyle\frac{8}{165} \) — Incorrect. This is only the case where the balls are white and he speaks the truth.
Advice for the exam: In the real exam, mark the answer that is mathematically correct. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.