What is the probability

Not too good, since the particles in the liquid medium is unstable.

The probability of getting a particle in a microliter is .006. If you pull 3000 microliters, you would expect a mean value of 3000 x .006 = 18 particles. The probability of getting N particles is:

(1 - .006) ^ (3000 - N) * .006 ^ N * 3000! / (3000 - N)! / N!

I got 7.9% chance for 15 particles.

I took statistics in college, but that was about five lifetimes ago.

It's my "thinking" that it should be a bell curve rather than a static number.

I could be wrong -

It's a discrete binomial function. It's like a mean time to failure problem. If something fails on average every 100 hours, what is the chance of making it 100 hours without a failure? In this case, think that every time you get a particle is the same as getting a failure, so if the probability of failure is 6 per 1000 microliters, how many failures do you expect in 3000 microliters?

The entire probability table:

Particles Probability
0 0.000%
1 0.000%
2 0.000%
3 0.001%
4 0.006%
5 0.023%
6 0.070%
7 0.182%
8 0.410%
9 0.823%
10 1.485%
11 2.437%
12 3.663%
13 5.083%
14 6.546%
15 7.865%
16 8.857%
17 9.385%
18 9.388%
19 8.894%
20 8.002%
21 6.854%
22 5.602%
23 4.378%
24 3.278%
25 2.356%
26 1.627%
27 1.082%
28 0.693%
29 0.429%
30 0.256%
31 0.148%
32 0.083%
33 0.045%
34 0.024%

I agree. A wee bit of googling seems to point to Poisson. I'm comfortable with probability, not so much with statistics. I couldn't think of a way to solve this with probability. The choice of 3000 is arbitrary, and different choices yield different (but close) results. I'm thinking engineer's solution is reasonable if instead of 300o drawings/samples, you calculate the limit as the number of drawings/samples approaches infinity.

The reason I was thinking about it were two-fold: 1) the water is always in flux, and 2) have you seen fishes grouped together in the ocean? They are rarely seen in even distribution.

Exactly. I chose 3000 as something close enough to infinity to get a good result and small enough to calculate without difficulty. That and a microliter worked well with the metric system.

I think the problem is intended to be about statistics and not about fluid mechanics or swarming behavior.

But it is about physics and natural phenomenon of nature.

Once you identified Poisson, I looked it up and there is a function in Excel that computes it. Excel reports back 0.078575525 for 15 when 18 is expected.

Nice! Just found this at wikipedia:

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.

Amazingly, that makes some sense to me! LOL

The usual model is that the number of particles in a 3 ml sample has Poisson distribution with parameter λ=(6)(3). If X is the number of particles in the sample, the model gives
\Pr(X=15)=e^{-18} \frac{(18)^{15}}{15!}.