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GATE 2018 CS – Question 10

General Aptitude · Quantitative Aptitude: Probability and Combinatorics · 2 marks · Multiple choice

A six sided unbiased die with four green faces and two red faces is rolled seven times. Which of the following combinations is the most likely outcome of the experiment?

  1. Three green faces and four red faces.
  2. Four green faces and three red faces.
  3. Five green faces and two red faces.
  4. Six green faces and one red face.

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Show answer and explanation

Correct answer: (C) Five green faces and two red faces.

Explanation

With $p(\text{green})=\frac23$, $P(k\text{ greens})=\binom7k(2/3)^k(1/3)^{7-k}\propto\binom7k2^k$: $k=3$: $35\cdot8=280$, $k=4$: $35\cdot16=560$, $k=5$: $21\cdot32=672$, $k=6$: $7\cdot64=448$. The most likely is five green and two red.