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Radioactive decay is a random process where unstable atoms lose energy by emitting radiation. Understanding this process is crucial in fields like nuclear engineering and medicine. Cesium-137, for instance, is a radioactive atom used in various applications, from cancer treatment to industrial gauges. Over time, cesium-137 releases its excess energy and becomes stable. In this exercise, we see that out of 1,000,000 atoms, 22,713 decayed over 365 days. This decay can be analyzed using the Poisson distribution, a probability distribution that describes the number of events happening within a fixed interval of time or space. It is particularly useful for rare events and can predict how frequently an atom will decay on any given day. Therefore, radioactive decay not only provides a practical example for learning Poisson distribution but also highlights real-world applications where understanding random events is essential.

The mean, or average, is a central concept in statistics and probability, indicating the average outcome of a random process. To calculate the mean in our exercise, we divide the total number of decayed atoms (22,713) by the number of days (365): \(\text{Mean} (\text{λ}) = \frac{22,713}{365} = 62.2 \text{ atoms/day} \). This mean value (\text{λ}) tells us that on average, 62.2 cesium-137 atoms decay each day. In Poisson distribution, this mean is used to predict the probability of a specific number of decays occurring on any given day. Calculating the mean simplifies our understanding, offering a single number that characterizes the typical daily decay rate. It serves as the foundation for further probabilistic calculations, allowing us to make informed predictions about random decay events.

Probability helps quantify the likelihood of an event occurring. In the context of Poisson distribution, we use the formula: \(\text{P(X = k)} = \frac{e^{-λ} \times λ^k}{k!}\) where \(\text{λ}\) is the mean number of events and \(\text{k}\) is the specific number of events for which we want to find the probability. For our exercise, we want to find the probability of exactly 50 atoms decaying in one day. Using the mean decay rate (\text{λ} = 62.2) and substituting \(\text{k} = 50\): \(\text{P(X = 50)} = \frac{e^{-62.2} \times 62.2^{50}}{50!}\) simplifies the process considerably. Through statistical calculators or software, we find that this probability is approximately \(\text{0.00414}\). Thus, there's a small yet quantifiable chance (around 0.41%) that exactly 50 of the cesium-137 atoms will decay on any given day. Understanding probability allows us to measure and predict the randomness inherent in natural processes, like radioactive decay, making it a powerful tool in both science and daily life.