Let N be the sum of the numbers obtained when two fair dice are rolled, and let the probab…

Question

Let $N$ be the sum of the numbers obtained when two fair dice are rolled, and let the probability that $N-2,\sqrt{3N},N+2$ are in G.P. be $\dfrac{k}{48}$. Then the value of $k$ is

Accepted Answer

G.P. condition: $3N=(N-2)(N+2)=N^2-4 \Rightarrow N^2-3N-4=0 \Rightarrow (N-4)(N+1)=0 \Rightarrow N=4$.

$n(N=4) = \{(1,3),(3,1),(2,2)\}=3$, out of 36 total.

$$P = \frac{3}{36}=\frac{1}{12}=\frac{4}{48} \implies k=4$$

Solutions · 1