Question

Three numbers are chosen at random from numbers 1 to 30. Write the probability that the chosen numbers are consecutive.

Answer 1

Let S be the sample space associated with three numbers that are chosen at random from the numbers 1 to 30.
∴ n(S) = ³⁰C₃ = $\frac{30!}{27!\times 3!}=\frac{30\times 29\times 28\times 27!}{27!\times 3\times 2\times 1}=5\times 29\times 28=4060$
Let E be the favourable number of elementary events that the chosen numbers are consecutive.
E = {(1, 2, 3), (2, 3, 4), (3, 4, 5),..., (27, 28, 29), (28, 29, 30)}
i.e. n(E) = ²⁸C₁ = 28
Hence, required probability = $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{28}{4060}=\frac{1}{145}$