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? A. Three green faces and four red faces. B. Four green faces and three red faces. C. Five green faces and two red faces. D. Six green faces and one red face.

GATE 2018 · General Aptitude · Probability · medium

Answer: The most likely outcome is five green faces and two red faces (option C), with probability 672/2187 ≈ 0.307.

1. Set up the binomial model: p = 2/3, 1-p = 1/3, n = 7. So P(X = k) = C(7,k) * (2/3)^k * (1/3)^(7-k). Common factor: 1/3^7 = 1/2187.
2. Compute numerators for each option: k=3: C(7,3)*2^3 = 35*8 = 280. k=4: C(7,4)*2^4 = 35*16 = 560. k=5: C(7,5)*2^5 = 21*32 = 672. k=6: C(7,6)*2^6 = 7*64 = 448.
3. Identify the most likely combination: Comparing: 280 < 448 < 560 < 672. Maximum is at k = 5 greens, 2 reds. So P(5 green, 2 red) = 672/2187 is the highest.