Question

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

Answer 1

$\because$ Three number are chosen from a number from 1-30

$\Rightarrow$ Number of elementary events in sample space is

$n\left(S\right){=}^{30}{C}_3=\frac{30!}{3!\times 27!}=\frac{30\times 29\times 28}{3\times 2\times 1}$

= $5\times 29\times 2=4060$

Let E be the event that three conecutive numbers are chosen

$n\left(E\right){=}^{28}{C}_1=28$

$\because$ E = {(1,2,3), (2,3,4), (3,4,5), (4,5,6), (5,6,7), ..... (27,28, 29), (28, 29, 30) }

$\Rightarrow n\left(E\right)=28$

$\therefore P\left(E\right)=\frac{28}{4060}=\frac{1}{145}$