Relationship between number density and probability

So for a homework problem for an astronomy class, we have a cluster of stars passing over a black hole, and need to find the probability that a star in the cluster will pass directly into the black hole's event horizon.

So, I found the number density of stars within the volume of intersection of the star cluster and event horizon.

I think the (mean) number density is almost the same thing as the probability in this case? But I'm not sure.

If the number density was 1 then there would on average be 1 star in the region, however I can't tell how to draw the probability from this.

If anyone could clear this up that would be helpful, thanks

EDIT

Ok, the question I asked is definitely unclear.

We approximate the star cluster as a sphere, and the black hole as a sphere. The radius of the black hole <<< the radius of the star cluster

Hence if the star cluster is moving, and 'passes over' the black hole, then the volume of intersection is approximated as a cylinder, with height 2 * star cluster radius, and the cross section of the cylinder has radius of the black hole.

We are told the total number of stars in the star cluster (0.5 million)

What i did was found the number density of the entire cluster, N / V, then assumed the density is uniform.

(I have just realised that I made a mistake in wording when i first wrote the question, which becomes clear next:)

Then i found the average number of stars N in the cylinder, by doing number density * cylinder volume.

So what I have found is the average number of stars in the region.

This number is ~10^(-9).

Is this equivalent the probability that a star lies in the region?

Your question isn't clear. I will answer one possible interpretation. If it's not what you mean then please edit the question to clarify.

If the stars are uniformly distributed in the cluster and the black hole is inside the cluster then the probability that a star in the cluster is in the black hole is just the ratio of the volume of the black hole to the volume of the cluster.

This probability is "per star" and is independent of the density of stars in the cluster (units stars per cubic meter). To find the number of stars in the black hole you just multiply the density, probability and volume of the black hole.

That density may or may not be what "number density" means.

I don't know what "passing over" means. It seems to suggest some kind of motion, but I see no data for that. The little I know about black holes makes leads me to believe that they may be attracting the stars in the cluster.

Edit in response to the question edit.

I think the calculation above works if you replace the volume of the black hole by the volume $$C$$ of the cylinder Then the probability that a star is swept up by the black hole is $$C/V$$, where $$V$$ is the volume of the spherical cluster. You don't need the number of stars or the density to find that probability, which will indeed be small. You do need it to find the number of stars swept up.

Notr an extra assumption: the black hole traverses the spherical cluster along a diameter. (I know you said it's the cluser that's moving, but the motion is relative ...)

• Because stars have mass, I suspect “number density” is a phrase used to mean average number of stars per volume and distinguish that from the mass density. – Steve Kass Nov 22 '18 at 0:02
• @SteveKass That makes sense to me. It is what I assumed. – Ethan Bolker Nov 22 '18 at 0:06
• I see, thank you for your time! – big daddy Nov 22 '18 at 0:17