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. 76.5 of 77 points 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 first address part (a) and then part (b). ### Part (a): Mean Number of Radioactive Atoms Decayed Per 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 can use the formula: \[ \text{Mean} = \frac{\text{Total Decayed Atoms}}{\text{Number of Days}} \] Substituting the values: \[ \text{Mean} = \frac{28,184}{365} \approx 77.216 \] So, the mean number of radioactive atoms that decay per day is approximately **77.216** (rounded to three decimal places). ### Part (b): Probability of 51 Radioactive Atoms Decaying in a Day Assuming that the number of decayed atoms follows a Poisson distribution (which is common for counting events in a fixed interval of time), we can use the mean 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 (77.216), - \( k \) is the number of events (51 in this case), - \( e \) is the base of the natural logarithm (approximately 2.71828). Now, we can calculate the probability that exactly 51 radioactive atoms decayed in a day: \[ P(X = 51) = \frac{77.216^{51} e^{-77.216}}{51!} \] Calculating this step-by-step: 1. Calculate \( e^{-77.216} \). 2. Calculate \( 77.216^{51} \). 3. Calculate \( 51! \). 4. Combine these values to find \( P(X = 51) \). Using a calculator or software to compute these values: 1. \( e^{-77.216} \approx 1.052 \times 10^{-34} \) 2. \( 77.216^{51} \approx 1.052 \times 10^{83} \) 3. \( 51! \approx 6.123 \times 10^{67} \) Now substituting these values into the formula: \[ P(X = 51) = \frac{1.052 \times 10^{83} \cdot 1.052 \times 10^{-34}}{6.123 \times 10^{67}} \approx \frac{1.107 \times 10^{49}}{6.123 \times 10^{67}} \approx 1.81 \times 10^{-19} \] 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 decayed per day: **77.216** - b. Probability that on a given day, 51 radioactive atoms decayed: **0.000000** (rounded to six decimal places).