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 problem states that the lifetime of a radioactive atom follows an exponential distribution with an average life span (mean) of 27 days. The probability density function for an exponential distribution is given by \( f(t) = \lambda e^{-\lambda t} \), where \( \lambda \) is the rate parameter. The rate parameter \( \lambda = \frac{1}{\text{mean}} = \frac{1}{27} \) per day.

Step 2: Calculate the Probability That Atom Will Not Decay in First 20 Days

The probability that the atom does not decay in the first 20 days is given by the survival function \( S(t) = e^{-\lambda t} \). For \( t = 20 \) days, we have:\[ S(20) = e^{-\frac{1}{27} \times 20} = e^{-\frac{20}{27}}. \]

Step 3: Compute the Specific Probability

Now we calculate the numeric value: \( e^{-\frac{20}{27}} \approx 0.478. \) Therefore, there is approximately a 47.8% chance that the atom does not decay in the first 20 days.

Step 4: Understand the Memoryless Property

The exponential distribution has a memoryless property, which means the probability of survival for the next 20 days given the atom has already survived the first 20 days is the same as the original survival probability for 20 days.

Step 5: Compute the Probability for the Next 20 Days Given the Atom Survived the First 20 Days

Using the memoryless property, the probability that the atom does not decay during the next 20 days, given it has survived the first 20 days, is also \( e^{-\frac{20}{27}} \approx 0.478. \)

Key Concepts

Probability Density Function

The probability density function (PDF) is a core concept of the exponential distribution. In this context, it's a mathematical function that describes how probabilities are distributed over possible outcomes. For an exponential distribution, the PDF is expressed as \( f(t) = \lambda e^{-\lambda t} \). Here, \( \lambda \) is called the rate parameter, which quantifies how often events occur. In our example of radioactive decay, this rate parameter \( \lambda \) is the reciprocal of the average lifespan, which means \( \lambda = \frac{1}{27} \) per day.

• \( t \) is the variable representing time.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• The function describes how the probability density spreads over time.

The PDF does not give probabilities directly, but you can integrate it over an interval to find the probability that an event falls within that interval. This makes it a fundamental tool for dealing with time until an event occurs, like radioactive decay.

Radioactive Decay

Radioactive decay is a naturally occurring process where unstable atomic nuclei lose energy by emitting radiation. The time taken for a radioactive substance to decay can be modeled using the exponential distribution, which provides a probabilistic framework to understand and predict decay behaviors. In the context of our example, the average time it takes for a radioactive atom to decay is known as its half-life, measured as 27 days. This 'memoryless' decay process means each moment holds the same potential for decay as the previous, often suited for situations where events do not depend on prior elapsed time.

• It applies not just to radioactive elements but to other contexts, such as electronic failures.
• The consistency of decay over time characterizes the exponential nature of the process.

Memoryless Property

One of the most interesting properties of the exponential distribution is its memoryless nature. This means that the probability of an event happening in the future is the same, regardless of how long you have already been waiting. In other words, the process 'forgets' how much time has elapsed. In our example, if we observe that a radioactive atom has survived the first 20 days, the likelihood it remains intact for another 20 days is exactly the same as if we were starting the observation at day 0.

• This property is unique to the exponential distribution.
• It simplifies calculations for ongoing probabilities over time.

Here, we use the survival function to express this, signifying consistent probabilities over fresh intervals, regardless of prior outcomes.

Survival Function

The survival function is a complementary concept to the probability density function, focusing more on 'survival' over time. It's a key tool in reliability analysis and risk assessment. For an exponential distribution, the survival function is expressed as \( S(t) = e^{-\lambda t} \), where \( \lambda \) is the rate parameter, and \( t \) represents time. The survival function gives the probability that an event has not occurred by a certain time \( t \). For our example involving a radioactive atom:\[ S(20) = e^{-\frac{20}{27}} \approx 0.478 \]

• This means there's a 47.8% chance the atom remains undecayed 20 days into observation.
• It's critical in systems where future reliability needs precise calculation.

This shows how well the exponential distribution models such decay scenarios, providing a clear picture of potential outcomes over distinct periods.