When studying radioactive material, a nuclear engineer found that over 365 days, \( 1,000,000 \) radioactive atoms decayed to 978,963 radioactive atoms, so 21,037 atoms decayed during 365 days. a. Find the mean number of radioactive atoms that decayed in a day. b. Find the probability that on a given day, 50 radioactive atoms decayed.

To solve the problem, we will first address part (a) and then part (b). ### Part (a): Mean Number of Radioactive Atoms Decayed in a Day To find the mean number of radioactive atoms that decayed in a day, we can use the total number of atoms that decayed over the entire period and divide it by the number of days. Given: - Total atoms decayed over 365 days = 21,037 atoms - Number of days = 365 The mean number of atoms decayed per day is calculated as follows: \[ \text{Mean} = \frac{\text{Total atoms decayed}}{\text{Number of days}} = \frac{21,037}{365} \approx 57.6 \] Thus, the mean number of radioactive atoms that decayed in a day is approximately **57.6 atoms**. ### Part (b): Probability that 50 Radioactive Atoms Decayed on a Given Day To find the probability that exactly 50 radioactive atoms decayed on a given day, we can model the decay of radioactive atoms using a Poisson distribution. The Poisson distribution is appropriate here because we are dealing with a count of events (decays) that occur independently over a fixed interval (a day). The Poisson distribution is defined as: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where: - \( \lambda \) is the mean number of events (in this case, the mean number of decays per day), - \( k \) is the number of events we want to find the probability for (in this case, 50 decays), - \( e \) is the base of the natural logarithm (approximately equal to 2.71828). From part (a), we found that \( \lambda \approx 57.6 \). Now, we can calculate the probability that exactly 50 radioactive atoms decayed in a day: \[ P(X = 50) = \frac{57.6^{50} e^{-57.6}}{50!} \] Calculating this directly can be complex due to the factorial and the exponentiation. However, we can use a calculator or software to compute this value. Using a calculator or software to compute \( P(X = 50) \): 1. Calculate \( 57.6^{50} \). 2. Calculate \( e^{-57.6} \). 3. Calculate \( 50! \). 4. Combine these values into the formula. After performing these calculations, we find: \[ P(X = 50) \approx 0.073 \] Thus, the probability that exactly 50 radioactive atoms decayed on a given day is approximately **0.073** or **7.3%**.