Ants on a tetrahedron

Probability Level 3

Four ants sit on the surface of a regular tetrahedron such that each ant is on a unique vertex. Simultaneously, each ant picks a new vertex distinct from the one they are currently on and travels to it with uniform speed.

If all ants travel at the same rate, what is the probability that none of the ants collided with one another at any time during this process?

1 27 \frac1{27} 1 81 \frac1{81} 8 27 \frac8{27} 1 9 \frac19 3 32 \frac3{32} 2 27 \frac2{27} 2 81 \frac2{81}

Jason Gonsalves by Jason Gonsalves

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1 solution

There are 3 4 3^4 possibilities. Now 1st ant can choose any vertex so 3 choices, 2nd ant cannot choose the 1st as they would collide midway so has 2 choices, 3rd ant cannot choose 2nd but also cannot choose 1st (as 4th ant would be left with no choices) and hence must move towards 4th ant which in turn can only move towards 1st .

0 Helpful 0 Interesting 0 Brilliant 2 Confused

I don't see why you say " 3rd ant cannot choose 2nd but also cannot choose 1st (as 4th ant would be left with no choices)". I think it is wrong, the third ant still has 2 choices: if he chooses 4th, then the 4th will go in place of the 1st, and if he chooses 1st, then 4th goes in place of him (the 3rd). Therefore the answer should be 4/27.

Nikita Mitin - 4 years, 7 months ago

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