First slide

Theorems of probability

Question

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral is

Moderate

A

$\frac{1}{2}$
B

$\frac{1}{5}$
C

$\frac{1}{10}$
D

$\frac{1}{20}$

Solution

Out of 6 vertices, 3 can be chosen $^{6} C_{3}$ , ways. The triangle will be equilateral if it is $Δ A C E$ or $Δ B D F$ (2 ways). Therefore, the required probability is
$\frac{2}{^{6} C_{3}} = \frac{2}{20} = \frac{1}{10}$

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