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 57.636 (Round to three decimal places as needed.) b. The probability that on a given day, 50 radioactive atoms decayed, is 0073 (Round to six decimal places as needed.)

To solve the problem, we will use the information provided to calculate the mean number of radioactive atoms that decayed per day and the probability of a specific number of atoms decaying on a given day. ### Part a: Mean Number of Radioactive Atoms Decayed Per Day We know that over 365 days, 21,037 atoms decayed. To find the mean number of atoms that decayed per day, we can use the formula: \[ \text{Mean} = \frac{\text{Total number of decayed atoms}}{\text{Number of days}} \] Substituting the values: \[ \text{Mean} = \frac{21,037}{365} \approx 57.636 \] So, the mean number of radioactive atoms that decayed in a day is approximately **57.636** (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 (which is common for counting events in a fixed interval of time), we can use the mean we calculated to find the probability of exactly 50 atoms decaying in a day. 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, 57.636), - \( k \) 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 2.71828). Substituting the values into the formula: \[ P(X = 50) = \frac{57.636^{50} e^{-57.636}}{50!} \] Calculating this requires evaluating \( 57.636^{50} \), \( e^{-57.636} \), and \( 50! \). Using a calculator or software to compute this: 1. Calculate \( 57.636^{50} \). 2. Calculate \( e^{-57.636} \). 3. Calculate \( 50! \). 4. Substitute these values into the formula. After performing these calculations, you should find: \[ P(X = 50) \approx 0.0073 \] Thus, the probability that on a given day, 50 radioactive atoms decayed is approximately **0.0073** (rounded to six decimal places).