Problem 85 - Calculus for Biology and Medicine

Question

Suppose the number of phone calls arriving at a switchboard per hour is Poisson distributed with mean 7 calls per hour. Find the probability that no phone calls arrive during a certain hour.

Step-by-Step Solution

Answer

The probability that no phone calls arrive during the hour is approximately 0.00091188.

Step 1: Understand the Question

This problem is asking for the probability that zero phone calls arrive in an hour when the number of calls follows a Poisson distribution with a mean (\( \lambda \)) of 7 calls per hour.

Step 2: Use the Poisson Probability Formula

The Poisson probability formula is given by: \( P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \) where \( k \) is the number of events (calls), \( \lambda \) is the mean number of events (7), and \( e \approx 2.71828 \) is the base of the natural logarithm. Here, \( k = 0 \).

Step 3: Substitute Values into the Formula

Using the formula, substitute \( \lambda = 7 \) and \( k = 0 \) to find \( P(X = 0) \):\[P(X = 0) = \frac{e^{-7} \cdot 7^0}{0!}\]

Step 4: Simplify the Expression

Simplify the expression for \( P(X = 0) \):- \( 7^0 = 1 \) and \( 0! = 1 \) since any number to the power of zero is 1 and the factorial of zero is 1.- Therefore, \( P(X = 0) = e^{-7} \).

Step 5: Calculate the Probability

Using a calculator or exponential table, compute \( e^{-7} \). The value is approximately 0.00091188.Therefore, \( P(X = 0) \approx 0.00091188 \).

Key Concepts

Probability Calculation

In probability theory, calculating the likelihood of a specific event is crucial, especially when working with statistical distributions like the Poisson distribution. Probability calculation helps us quantify how likely it is for an event to occur. For a Poisson distribution, the probability that exactly \( k \) events happen in a set interval of time or space is determined by the formula:

• \( P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \)
• Where:
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828
• \( \lambda \) is the average number of events
• \( k \) is the number of events for which we want to calculate the probability

Finding the probability involves substituting the known values into the formula and simplifying. This systematic approach ensures we understand both the process and the logic behind the calculation.
For example, in our original problem, we wanted to find the probability of receiving zero calls (\( k = 0 \)) in one hour when the mean number of calls is 7 (\( \lambda = 7 \)). By substituting these values into the formula, we determined that \( P(X = 0) \approx 0.00091188 \). This means there is a very small chance that no calls arrive within that period, supported by the computed value.

Mean Number of Events

The mean number of events, often represented as \( \lambda \), is a critical parameter in the Poisson distribution. This number tells us the average occurrence of an event within a fixed interval, like time or space. In the context of phone calls arriving at a switchboard, \( \lambda = 7 \) means on average, 7 calls arrive per hour.

• It is important to remember that \( \lambda \) does not dictate the exact number of events that will occur but instead offers a central tendency or expected value.
• The Poisson distribution is particularly useful when these events occur independently and do not influence each other.

Understanding \( \lambda \) helps us make informed predictions about the variability of an event. It allows us to use the Poisson probability formula to calculate the chances of different outcomes.
In our problem, knowing \( \lambda = 7 \) guided us in predicting the seemingly unlikely outcome of receiving zero calls, quantifying it via the appropriate probability formula.

Exponential Function

The exponential function is a mathematical function denoted by \( e^x \), where \( e \) is approximately 2.71828, and it is the base of the natural logarithm. This function plays a pivotal role in Poisson distributions because it handles the decay of probability as the number of mean events increases.

• In a Poisson distribution, it is embedded in the formula: \( P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \).
• The function \( e^{-\lambda} \) represents the probability that fewer than \( k \) average events occur.

In our specific example, calculating \( e^{-7} \) helps determine the drop-off probability that zero events happen, a crucial step in accurately solving our problem. The outcome of \( e^{-7} \) is approximately 0.00091188, showcasing the vital impact the exponential function has in determining low-probability events over a large mean value.
By understanding and using the exponential function, we can spread our comprehension over various probability scenarios, making it a powerful tool for statistical analysis.