When studying radioactive material, a nucloar engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,307 radioactive atoms, so 22,693 atoms decayed during 365 days. a. Find the mean number of radioactive atoms that docayed 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 62.173 (Round to three decimal places as needed.) b. The probability that on a given day, 50 radioactive atoms decayed, is (Round to six decimal places as needod.)

To solve the problem, we will use the information provided to find the mean number of radioactive atoms that decay per day and then calculate the probability of exactly 50 atoms decaying on a given day using the Poisson distribution. ### Part a: Mean Number of Radioactive Atoms Decayed Per Day We know that over 365 days, 22,693 atoms decayed. To find the mean number of atoms that decay per day, we can use the formula: \[ \text{Mean} = \frac{\text{Total Decayed Atoms}}{\text{Number of Days}} = \frac{22,693}{365} \] Calculating this: \[ \text{Mean} \approx 62.173 \] So, the mean number of radioactive atoms that decay per day is approximately **62.173** (rounded to three decimal places). ### Part b: Probability of Exactly 50 Atoms Decaying in a Day To find the probability of exactly 50 atoms decaying in a day, we can use the Poisson distribution, which is appropriate for modeling the number of events (in this case, decays) in a fixed interval of time when these events occur with a known constant mean rate and independently of the time since the last event. The probability mass function of the Poisson distribution is given by: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where: - \( \lambda \) is the mean number of occurrences (62.173 in this case), - \( k \) is the number of occurrences we want to find the probability for (50), - \( e \) is the base of the natural logarithm (approximately 2.71828). Substituting the values into the formula: \[ P(X = 50) = \frac{62.173^{50} e^{-62.173}}{50!} \] Calculating this requires evaluating \( 62.173^{50} \), \( e^{-62.173} \), and \( 50! \). Using a calculator or software to compute this: 1. Calculate \( e^{-62.173} \). 2. Calculate \( 62.173^{50} \). 3. Calculate \( 50! \). 4. Substitute these values into the formula. After performing these calculations, you should find: \[ P(X = 50) \approx 0.051052 \] Thus, the probability that on a given day, exactly 50 radioactive atoms decayed is approximately **0.051052** (rounded to six decimal places).