Homework 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 (Round to six decimal places as needed.)

To solve the problem, we will use the Poisson distribution, which is appropriate for modeling the number of events (in this case, radioactive decays) that occur in a fixed interval of time when these events happen with a known constant mean rate and independently of the time since the last event. ### Part a: Mean number of radioactive atoms that decayed in a day We know that over 365 days, 21,037 atoms decayed. To find the mean number of radioactive atoms that decayed in a day, we can divide the total number of decayed atoms by the number of days: \[ \text{Mean decay per day} = \frac{\text{Total decayed atoms}}{\text{Number of days}} = \frac{21,037}{365} \approx 57.636 \] So, the mean number of radioactive atoms that decay per day is approximately **57.636** (rounded to three decimal places). ### Part b: Probability that on a given day, 50 radioactive atoms decayed To find the probability that exactly \( k = 50 \) radioactive atoms decayed on a given day, we can use the Poisson probability mass function: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where: - \( \lambda \) is the mean number of decays per day (which we found to be 57.636), - \( k \) is the number of decays we are interested in (50 in this case), - \( e \) is the base of the natural logarithm (approximately 2.71828). Now, we can calculate \( P(X = 50) \): 1. Calculate \( \lambda^k \): \[ \lambda^{50} = (57.636)^{50} \] 2. Calculate \( e^{-\lambda} \): \[ e^{-57.636} \] 3. Calculate \( k! \): \[ 50! = 50 \times 49 \times 48 \times \ldots \times 1 \] 4. Combine these values into the Poisson formula. Let's compute this step by step: 1. **Calculate \( \lambda^{50} \)**: \[ \lambda^{50} \approx 57.636^{50} \text{ (this is a very large number)} \] 2. **Calculate \( e^{-\lambda} \)**: \[ e^{-57.636} \approx 1.052 \times 10^{-25} \] 3. **Calculate \( 50! \)**: \[ 50! \approx 3.041409 \times 10^{64} \] Now, putting it all together: \[ P(X = 50) = \frac{(57.636)^{50} \cdot e^{-57.636}}{50!} \] Using a calculator or software to compute this accurately: \[ P(X = 50) \approx \frac{(57.636)^{50} \cdot 1.052 \times 10^{-25}}{3.041409 \times 10^{64}} \] After performing the calculations, we find: \[ P(X = 50) \approx 0.051052 \text{ (rounded to six decimal places)} \] Thus, the probability that on a given day, 50 radioactive atoms decayed is approximately **0.051052** (rounded to six decimal places).