Question

Probability of at least one encounter Suppose in a large group of people, a fraction \(0 \leq r \leq 1\) of the people have flu. The probability that in \(n\) random encounters you will meet at least one person with flu is \(P=f(n, r)=1-(1-r)^{n} .\) Although \(n\) is a positive integer, regard it as a positive real number. a. Compute \(f_{r}\) and \(f_{n^{*}}\) b. How sensitive is the probability \(P\) to the flu rate \(r ?\) Suppose you meet \(n=20\) people. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.1\) to \(r=0.11(\text { with } n \text { fixed }) ?\) c. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.9\) to \(r=0.91\) with \(n=20 ?\) d. Interpret the results of parts (b) and (c).

Short answer

#tag_title# Compute \(f_r\) for \(n=20\) #tag_content#First, let's compute the partial derivative \(f_r\) for \(n=20\): \(f_r(20, r) = 20(1-r)^{20-1}\) #tag_title# Compute the change in probability #tag_content#When the flu rate increases from \(r=0.1\) to \(r=0.11\), the change in probability can be approximated by multiplying the derivative \(f_r\) evaluated at \(r=0.1\) by the change in \(r\) (which is \(0.01\)). Thus, the change in probability is: \(\Delta P \approx f_r(20, 0.1) \cdot 0.01 = 20(1-0.1)^{19} \cdot 0.01 \approx 0.028\) So, approximately, the probability \(P\) increases by 0.028 when the flu rate increases from \(0.1\) to \(0.11\) with \(n\) fixed at 20. #Interpretation of the result The sensitivity of the probability \(P\) to the flu rate \(r\) indicates how much the probability of meeting at least one person with the flu changes as the flu rate changes. In this case, we can see that as the flu rate increases by a small amount (from \(0.1\) to \(0.11\)), the probability \(P\) increases by approximately 0.028. This indicates that the risk of meeting at least one person with the flu increases as the flu rate increases, even if the increase in the flu rate is relatively small.

Step-by-step solution

Compute \(f_r\)

To compute the partial derivative of \(f(n, r)\) with respect to \(r\), we can treat \(n\) as a constant and differentiate \(f(n, r)\) with respect to \(r\). The partial derivative \(\frac{\partial f}{\partial r}\) can be computed as follows: \(\frac{\partial f}{\partial r} = \frac{\partial}{\partial r}\left[1 - (1-r)^n\right] = n(1-r)^{n-1}\)

Compute \(f_{n^*}\)

To compute the partial derivative of \(f(n, r)\) with respect to \(n\), we need to differentiate \(f(n, r)\) with respect to \(n\) by treating \(r\) as a constant. We can use the chain rule to compute \(\frac{\partial f}{\partial n}\) as follows: \(\frac{\partial f}{\partial n} = \frac{\partial}{\partial n}\left[1 - (1-r)^n\right] = -(1-r)^n \ln(1-r)\) #b. How sensitive is the probability \(P\) to the flu rate \(r ?\) Suppose you meet \(n=20\) people. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.1\) to \(r=0.11(\text { with } n \text { fixed }) ?\) To determine how sensitive the probability \(P\) is to the flu rate \(r\), we can compute \(f_r\) for \(n=20\) and then approximate the change in probability when \(r\) increases from \(0.1\) to \(0.11\).

Key concepts

Partial Derivative

Understanding the concept of a partial derivative is fundamental when dealing with functions of multiple variables. Imagine you're in a garden with beautiful flowers representing our function, and you only water one type of flower at a time to see its growth - that's the essence of a partial derivative. It tells you how much the function changes when you alter just one variable, keeping all others fixed.

In the context of our exercise, with the probability function given by P = f(n, r) = 1 - (1 - r)^n, the partial derivative with respect to r, denoted as f_r, is like asking: If the flu rate changes ever so slightly, but we meet the exact same number of people, how does the probability of encountering at least one person with flu change? We calculate f_r and in this case, the growth (or decrease) of our probability 'flower' is given by n(1 - r)^(n-1). This gives us a fascinating glimpse into the interconnectedness of variables within our function.

Sensitivity to Flu Rate

Let's dive into the sensitivity of our probability to changes in the flu rate. Sensitivity, in simple terms, is a measure of how responsive the probability of an encounter with a flu carrier is to changes in the flu rate - a bit like checking how sensitive your skin is to sunlight.

Using our partial derivative f_r, we get a sense of this responsiveness. With n fixed at 20, we see how a small increase in r from 0.1 to 0.11 shifts our probability. This is akin to evaluating whether a slight uptick in flu cases has a big or small impact on the chance of bumping into someone who's unwell. We apply the concept of partial derivatives to get a numerical estimate of this sensitivity.

Probability Increase Estimation

Estimating the increase in probability helps us to forecast the risk of certain events. In a way, it's like predicting the chance of rain based on dark clouds. In the exercise, we are asked to estimate how much more likely it is to meet someone with the flu if the flu rate rises marginally, keeping the number of encounters constant.

Using the sensitivity analysis from our partial derivative, we calculate this by multiplying the change in flu rate Δr by f_r. It's a prediction, not an exact value, like guessing how much heavier rain might be when those clouds get darker. It provides us with a useful approximation on how a small change in one factor can alter the outcome significantly or just a little bit.

Chain Rule

The chain rule is a powerful tool in calculus, essential for crunching the numbers on more complicated functions. Imagine it as a domino effect - you nudge the first block (the outer function), which in turn nudges the next (the inner function), and the result is the combined effect of all these nudges on our function's value.

In our problem, we apply the chain rule to differentiate f(n, r) with respect to n. This shows us what happens to the probability when we increment the number of encounters while keeping the flu rate constant. We're essentially evaluating how changing one aspect of the situation - the number of people we're meeting - could alter the overall outcome. And just like that, using the chain rule, we uncover the ripple effect of one variable on our probability function.