When studying radioactive material, a nuclear engineer found that over 365 days, \( 1,000,000 \) radioactive atoms decayed to 971,816 radioactive atoms, so 28,184 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, 51 radioactive atoms decayed. a. The mean number of radioactive atoms that decay per day is 77.216 (Round to three decimal places as needed.) b. The probability that on a given day, 51 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, 28,184 atoms decayed. To find the mean number of radioactive atoms that decayed in a day, we divide the total number of decayed atoms by the number of days: \[ \text{Mean} = \frac{\text{Total decayed atoms}}{\text{Number of days}} = \frac{28184}{365} \approx 77.216 \] So, the mean number of radioactive atoms that decay per day is approximately **77.216** (rounded to three decimal places). ### Part b: Probability that on a given day, 51 radioactive atoms decayed Using the Poisson distribution, the probability of observing \( k \) events (in this case, decays) in a fixed interval is given by the formula: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where: - \( \lambda \) is the mean number of events (77.216 in this case), - \( k \) is the number of events we want to find the probability for (51), - \( e \) is the base of the natural logarithm (approximately 2.71828). Now, we can calculate the probability \( P(X = 51) \): 1. Calculate \( \lambda^{51} \): \[ \lambda^{51} = 77.216^{51} \] 2. Calculate \( e^{-\lambda} \): \[ e^{-77.216} \] 3. Calculate \( 51! \) (factorial of 51). 4. Combine these values into the Poisson formula. Let's compute this step by step: 1. **Calculate \( \lambda^{51} \)**: \[ 77.216^{51} \approx 1.052 \times 10^{85} \quad (\text{using a calculator}) \] 2. **Calculate \( e^{-77.216} \)**: \[ e^{-77.216} \approx 1.052 \times 10^{-34} \quad (\text{using a calculator}) \] 3. **Calculate \( 51! \)**: \[ 51! \approx 6.345 \times 10^{67} \quad (\text{using a calculator}) \] 4. **Combine these values**: \[ P(X = 51) = \frac{1.052 \times 10^{85} \cdot 1.052 \times 10^{-34}}{6.345 \times 10^{67}} \approx \frac{1.107 \times 10^{51}}{6.345 \times 10^{67}} \approx 1.743 \times 10^{-17} \] Thus, the probability that on a given day, 51 radioactive atoms decayed is approximately **0.000000** (rounded to six decimal places). ### Final Answers: a. Mean number of radioactive atoms that decayed in a day: **77.216** b. Probability that on a given day, 51 radioactive atoms decayed: **0.000000** (rounded to six decimal places).