Final answer:
Using the Poisson distribution, the fraction of plates with exactly 3 colonies is approximately 22.4%, with the assumption that the average number of colonies per plate is 3. This answer utilizes a common statistical distribution often applied to model events occurring at a constant rate within a fixed interval.
Step-by-step explanation:
The student's question pertains to the occurrence of colonies in a microbiological experiment. In the context of Airborne spores producing tiny mold colonies on gelatin plates, the question asks for the fraction of plates that has exactly 3 colonies given an average of 3 colonies per plate and then extends the question to ask about the fraction of plates with exactly m colonies, posited as a large integer.
Since the exact distribution of colonies on plates is not specified, we can assume a Poisson distribution because it is often used to model the number of times an event occurs within a given unit of time or in a given space. For the Poisson distribution, when the mean (average number of occurrences) is m, the probability (fraction of plates) having exactly m occurrences (colonies) is given by the formula:
P(X=m) = (e-m * mm) / m!
Where e is the base of the natural logarithm, approximately equal to 2.71828, m is the average number of colonies, and m! is the factorial of m.
Therefore, applying the formula for the given average of 3 colonies per plate, we compute:
P(X=3) = (e-3 * 33) / 3! = (0.04979 * 27) / 6 approximately equals to 0.224 or 22.4%
This means that about 22.4% of the plates are expected to have exactly 3 mold colonies.