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. a. The mean number of radioactive atoms that decay per day is 57636 (Round to three decimal places as needed.) b. The probability that on a given day, 50 radioactive atoms decayed, is 00.073 (Round to six decimal places as needed.)

To solve the problem, we will first address part (a) and then part (b). ### Part (a): Mean Number of Radioactive Atoms Decayed Per Day The total number of radioactive atoms that decayed over 365 days is given as 21,037. To find the mean number of atoms that decayed per day, we divide the total number of decayed atoms by the number of days: \[ \text{Mean decay per day} = \frac{\text{Total decayed atoms}}{\text{Number of days}} = \frac{21,037}{365} \] Calculating this: \[ \text{Mean decay per day} \approx 57.6 \] So, the mean number of radioactive atoms that decayed in a day is approximately **57.6** (rounded to three decimal places). ### Part (b): Probability of 50 Radioactive Atoms Decaying in a Day Assuming that the decay of radioactive atoms follows a Poisson distribution, we can use the mean decay rate we calculated in part (a) as the parameter \( \lambda \) of the Poisson distribution. The probability mass function of a Poisson distribution is given by: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where: - \( \lambda \) is the mean number of occurrences (in this case, the mean decay per day), - \( k \) is the number of occurrences we want to find the probability for (in this case, 50), - \( e \) is the base of the natural logarithm (approximately equal to 2.71828). Using \( \lambda \approx 57.6 \) and \( k = 50 \): \[ P(X = 50) = \frac{57.6^{50} e^{-57.6}}{50!} \] Calculating this requires evaluating \( 57.6^{50} \), \( e^{-57.6} \), and \( 50! \). This can be quite complex, so it's often easier to use a calculator or software that can handle these calculations. After performing the calculations, you would find: \[ P(X = 50) \approx 0.073 \] Thus, the probability that on a given day, 50 radioactive atoms decayed is approximately **0.073** (rounded to six decimal places). ### Summary of Results - **Mean number of radioactive atoms decayed per day**: 57.6 - **Probability that on a given day, 50 radioactive atoms decayed**: 0.073