Ants on polygon

Probability Level 3

There are 20 ants on different vertices of a 20-sided polygon. What is the probability of collision (between any two or all of them) if they start walking on the sides of the polygon?

If this probability is of the form ${a - \dfrac{b}{c} }$ , where $a,b,c$ are all positive integers with $a,b,c$ are all positive integers with $(b,c)$ coprime and $b< c$ , enter your answer as $a+b+c$ .

The answer is 524290.

2 solutions

Ossama Ismail
Jan 9, 2017

There are $2^{20}$ ways they can move. there are only $\color{#D61F06} 2$ ways in which the ants can move to avoid any collision clockwise or counter clockwise.Therefore, probability of collision is $(2^{20} - {\color{#D61F06} 2 }) / 2^{20} = 1 - 1/2^{19} = a - \frac{b}{c}$ .

Answer $= 1 + 1 + 524288 = 524290$

Better you should mention that all ants are different or numbered and specify what collision between all ants mean.

Vishal Yadav - 4 years, 3 months ago

A Former Brilliant Member
Jan 9, 2017

For any polygon of $n$ ants and $n$ vertices, the probability of collision is $1-\frac{2}{2^n}$ $=1-\frac{2}{2^{20}}=1-\frac{1}{524288}$

$a+b+c=1+152488=524290$

Ants on polygon

The answer is 524290.

2 solutions

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