Seeing Red

Probability Level 3

An initially white octahedron has four of its sides randomly painted red.

What is the probability that you can orient the octahedron so that when one vertex points toward you (as in the above picture) all you can see are red sides?

If the probability is $\dfrac{a}{b}$ where $a$ and $b$ are coprime positive integers, express your answer as $a+b$ .

Image credit: http://web.eecs.utk.edu

The answer is 38.

1 solution

Geoff Pilling
Oct 18, 2018

For a given orientation there are $\binom{8}{4}=70$ ways the sides can be painted.

Once painted, $6$ of those 70 possible ways allow us to reorient the octahedron to look like the picture - all red when looking down one vertex (one for clustering the reds around each vertex).

Therefore the probability is $\dfrac{6}{70} = \dfrac{3}{35}$

$3 + 35 = \boxed{38}$

The question asks the probability of orienting the octahedron such that we see the red sides, but your solution gives the value for a given orientation. I am new to probability so I am confused as to why these two mean the same thing.

Abha Vishwakarma - 2 years, 7 months ago

Good point... The solution was a little misleading. Does it make more sense now? I've reworded it a bit.

Geoff Pilling - 2 years, 7 months ago

Yeah, now I get it. Thanks.

Abha Vishwakarma - 2 years, 7 months ago

Seeing Red

The answer is 38.

1 solution

0 pending reports