Problem 51 - Calculus for Biology and Medicine

Question

Suppose that the lifetime of a radioactive atom is exponentially distributed with an average life span of 27 days. (a) Find the probability that the atom will not decay during the first 20 days after you start to observe it. (b) Suppose that the atom does not decay during the first 20 days that you observe it. What is the probability that it will not decay during the next 20 days?

Step-by-Step Solution

Answer

(a) 0.478; (b) 0.478.

Step 1: Understand the Exponential Distribution

The lifetime of the radioactive atom follows an exponential distribution. The probability density function (PDF) for an exponential distribution is given by \( f(t) = \frac{1}{\lambda} e^{-\frac{t}{\lambda}} \), where \( \lambda \) is the mean lifetime. For this problem, \( \lambda = 27 \) days.

Step 2: Calculate the Rate Parameter

The rate parameter \( \beta \) for the exponential distribution is the reciprocal of the mean. Thus, \( \beta = \frac{1}{\lambda} = \frac{1}{27} \).

Step 3: Find the Probability of Surviving 20 Days

Using the exponential distribution survival function \( S(t) = e^{-\beta t} \), where \( t \) is 20 days: \( S(20) = e^{-\frac{20}{27}} \).

Step 4: Calculate \( S(20) \) Numerically

Substitute the values into the expression: \( S(20) = e^{-\frac{20}{27}} \approx 0.478 \). Hence, the probability that the atom will not decay in 20 days is approximately 0.478.

Step 5: Understand the Memoryless Property

An important characteristic of the exponential distribution is the memoryless property. This means that given the atom has not decayed in the first 20 days, the probability distribution of the remaining lifetime is still exponential with rate \( \beta \).

Step 6: Find the Probability of Surviving Another 20 Days

Given the atom has not decayed during the first 20 days, calculate the probability it survives another 20 days: \( S(20) = e^{-\frac{20}{27}} \approx 0.478 \). The memoryless property ensures that this is the same as calculated in step 3.

Key Concepts

Radioactive Decay

Radioactive decay is a process by which unstable forms of atoms lose their energy and transform into more stable ones, typically releasing particles or gamma rays in the process. This decay occurs at a random rate, making it unpredictable on a single-atom basis. Rather than trying to forecast the exact point an atom will decay, we calculate these events' likelihood using probability distributions. For atoms, the exponential distribution is often used as it naturally models situations where the probability of an event (like decay) is constant over time. By representing lifetime as an exponential distribution, we can understand not just the mean or average life span but also derive the chances of a radioactive atom surviving a certain period without decaying. This concept is crucial in fields like nuclear physics, helping to predict the behavior of radioactive substances.

Probability Calculation

Understanding the exponential distribution helps us calculate important probabilities, such as the decay of radioactive atoms over time. The calculation is based on the exponential probability density function (PDF), given by \[ f(t) = \frac{1}{\lambda} e^{-\frac{t}{\lambda}} \]where \( \lambda \) represents the average lifecycle. In problems like these, the probability we want focuses on how long the atom can survive without decaying, expressed through the survival function:\[ S(t) = e^{-\beta t} \]Here, \( \beta \) is the rate parameter (\( \beta = \frac{1}{\lambda} \)), which inversely corresponds to the mean. For instance, if the mean lifetime is 27 days, we find \( \beta = \frac{1}{27} \). By substituting time \( t \) into the survival function, we get the probability of surviving, say, 20 days: \[ S(20) = e^{-\frac{20}{27}} \approx 0.478 \]This indicates a roughly 47.8% chance that the atom does not decay in the first 20 days since observation started.

Memoryless Property

A fascinating and unique feature of the exponential distribution is its memoryless property. This concept means that the probability of an atom surviving additional time given it has already survived some amount of time is the same as its original survival probability. In essence, it "forgets" how much time has passed.

Let's consider this with an example: If an atom has not decayed during the first 20 days, the probability of it not decaying in the next 20 days is still about 47.8%, similar to its survival probability for the first 20 days. The formula remains consistent because the exponential distribution resets itself, regardless of prior outcomes. So, whether we calculate the probability at day 0 or day 20 makes no difference for future increments of time. This reset mechanism is a compelling reason why the exponential distribution accurately models many real-world random processes, where history doesn't influence the current probability assessment.