A fair six-faced die, numbered 1 to 6, is rolled four times. What is the probability that the number 6 comes up exactly twice?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\frac{25}{216}\)
Four rolls give \(6\times6\times6\times6=1296\) equally likely outcomes. For exactly two sixes, choose which 2 rolls show 6; each of the other two rolls must show one of the other 5 numbers.
Ways to choose the 2 rolls showing 6 \(\displaystyle ={}^{4}C_2=\frac{4\times3}{2}=6\)
The other two rolls show 1, 2, 3, 4 or 5: \(5\times5=25\) ways
Favourable outcomes \(=6\times25=150\)
Probability \(\displaystyle =\frac{150}{1296}=\frac{25}{216}\)