Question

If a drop of water is placed on a slide and examined under a microscope, the number \(x\) of a specific type of bacteria present has been found to have a Poisson probability distribution. Suppose the maximum permissible count per water specimen for this type of bacteria is five. If the mean count for your water supply is two and you test a single specimen, is it likely that the count will exceed the maximum permissible count? Explain.

Short answer

Answer: No, it is not likely. The probability of the count exceeding the maximum permissible count is 1.66%, which is quite low.

Step-by-step solution

Understand Poisson Distribution

The Poisson distribution is used to model the number of events happening in a fixed interval of time or space, given the average rate of events. In our case, we have the Poisson distribution to model the number of bacteria present in the water specimen.

Identify the Parameters

We are given that the mean count (average number) of bacteria in the water specimen is 2. Hence, λ (lambda) equals 2. We want to find the probability of getting more than 5 bacteria in the sample.

Calculate the Probability

We want to find the probability \(P(X > 5)\). Instead, we calculate the probability that it is equal or lower and then subtract it from 1. So, we need to find \(P(X \leq 5) = P(X = 0) + P(X = 1) + ... + P(X = 5)\) where X denotes the number of bacteria in the sample. Using the formula for Poisson distribution, \(P(X=k) = \frac{e^{-λ} λ^k}{k!}\). So, by plugging in values for each k, we get: $P(X = 0) = \frac {e^ {-2} 2^0} {0!} \\ P(X = 1) = \frac {e^ {-2} 2^1} {1!} \\ P(X = 2) = \frac {e^ {-2} 2^2} {2!} \\ P(X = 3) = \frac {e^ {-2} 2^3} {3!} \\ P(X = 4) = \frac {e^ {-2} 2^4} {4!} \\ P(X = 5) = \frac {e^ {-2} 2^5} {5!}$ We can then sum the above probabilities: \(P(X \leq 5) = \sum_{k=0}^5 \frac {e^ {-2} 2^k} {k!}\)

Calculate the Probability of Count Exceeding 5

To calculate the probability of the count exceeding five, we can subtract the probability of getting less than or equal to five from 1: \(P(X>5) = 1 - P(X\leq5)\) Using a calculator, we find \(P(X \leq 5) \approx 0.9834\) So, \(P(X > 5) = 1 - 0.9834 = 0.0166\)

Make a Conclusion

The probability that the count will exceed the maximum permissible count is 0.0166 or 1.66%. This probability is quite low. So, it is not likely that the count will exceed the maximum permissible count if you test a single water specimen from your water supply.

Key concepts

Probability Calculation

Understanding how to calculate probability with the Poisson distribution is crucial when dealing with scenarios where you expect a certain number of events over a fixed period or in a fixed space. This exercise asks us about the likelihood that the count of a specific type of bacteria exceeds a threshold number in a water sample. For a Poisson distribution, the probability of observing a precise number of events, say \(X\) events, is calculated using the formula: - \(P(X=k) = \frac{e^{-λ} λ^k}{k!}\) where \(λ\) (lambda) is the average rate of events (the mean number of occurrences), \(e\) is approximately 2.71828 (Euler's number), \(k\) is the number of occurrences we want to find the probability for, and \(!\) denotes factorial, which is the product of all positive integers up to that number.To find the probability of having up to 5 bacteria, we calculate each probability from \(P(X = 0)\) to \(P(X = 5)\) and sum them. This gives us \(P(X \leq 5)\). However, if we are interested in probabilities like exceeding a certain number, we compute \(P(X > 5)\) by subtracting the sum of these probabilities from 1. This method allows us to derive probabilities for scenarios beyond specific counts.

Mean Rate of Events

The mean rate of events, denoted as \(λ\) in the Poisson distribution, is a foundational concept that represents the average number of occurrences in a given interval. In our exercise, this mean rate is given as 2, which means on average, 2 bacteria are expected to be observed under the microscope per water specimen. This number is not just a central value, but a parameter that shapes the distribution and thus helps us determine the likelihood of different counts of bacteria.The significance of the mean rate in the Poisson model is that it dictates how often an event, like observing the bacteria, is expected to occur. By understanding this rate, one can apply it to predict probabilities for various counts:

• If the mean rate is low, we generally expect fewer events in the interval, and thus lower probabilities for higher counts.
• Conversely, a higher mean rate would suggest more frequent occurrences of the event within the same interval, impacting probabilities towards trending higher counts.

This understanding allows effective modeling of real-world phenomena where events occur at a constant average rate.

Probability of Exceeding

With Poisson distributions, sometimes the focus is on not just the exact counts but whether they exceed a certain threshold. This becomes useful for risk assessment or quality control, such as determining if safety limits are breached in our example.In the given problem, the aim is to find out if the count of bacteria exceeds 5 using the known average of 2. After calculating \(P(X \leq 5)\) as about 0.9834, we derive \(P(X > 5)\) by subtracting from 1, arriving at approximately 0.0166 or 1.66%.This probability tells us:

• The chances that any single sample exceeds the permissible maximum count are very low.
• It helps in making informed decisions about whether further actions or improvements need to be considered, based on the likelihood of breaching the safety count.

Thus, understanding whether an event exceeds a certain threshold gives insights into the potential for exception handling or alerts for attention when the risk is unacceptably high.