If an unbiased dice is rolled thrice, then the probability of getting a greater number in the i^{th}

Question

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \(i^{\text {th }}\) roll than the number obtained in the\( (i-1)^{\text {th }}\) roll, \(\mathrm{i}=2,3\), is equal to :

(1) \(3 / 54\)

(2) \(2 / 54\)

(3) \(5 / 54\)

(4) \(1 / 54\) 

Answer 1

Correct option is : (3) 5/54 

Let \(\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3\) be the numbers on \(1^{\text {st }}, 2^{\text {nd }} \ \&\ 3^{\mathrm{rd}}\) throw respectively.

Given:  \(\mathrm{x}_1<\mathrm{x}_2<\mathrm{x}_3\) 

\(\therefore\) Favorable cases \(={ }^6 \mathrm{C}_3\)

Total cases \( =6^3=216\) 

\( \text { Prob }=\frac{{ }^6 \mathrm{C}_3}{216} \Rightarrow \frac{20}{216}=\frac{5}{54} \)