Three number are chosen at random from number 1 to 30. Write the … - Vidyadip

Question

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

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Answer

$\because$ Three number are choosen from a number from 1-30
⇒ Number of elementary events in sample space is
$\text{n}(\text{S})=^{30}\text{C}_3=\frac{30\times29\times28}{3\times2\times1}$
$=5\times29\times28$
$=4060$
Let E be the event that three consecutive number are choosen
$\text{n}(\text{E})=^{28}\text{C}_1=28$
$\because\text{E}=\big\{(1,\ 2,\ 3),\ (2,\ 3,\ 4),\ (3,\ 4,\ 5),\ (4,\ 5,\ 6),\\ \ \ \ (5,\ 6,\ 7),\ .....(27,\ 28,\ 29),\ (28,\ 29,\ 30)\big\}$
$\Rightarrow\text{n}(\text{E})=28$
$\therefore\text{p}(\text{E})=\frac{28}{4060}$
$=\frac{1}{145}$

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