When studying radioactive material, a nuclear engineer found that over 365 days, \( 1,000,000 \) radioactive atoms decayed to 979,277 radioactive atoms, so 20,723 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, 53 radioactive atoms decayed. a. The mean number of radioactive atoms that decay per day is 56.775 (Round to three decimal places as needed.) b. The probability that on a given day, 53 radioactive atoms decayed, is (Round to six decimal places as needed )

a. To find the mean number of radioactive atoms that decayed in a day, we can use the formula for the mean: Mean = Total number of atoms decayed / Total number of days Given that 20,723 atoms decayed over 365 days, we can calculate the mean as follows: Mean = 20,723 / 365 Let's calculate the mean. Calculate the value by following steps: - step0: Calculate: \(\frac{20723}{365}\) The mean number of radioactive atoms that decayed in a day is approximately 56.775. b. To find the probability that on a given day, 53 radioactive atoms decayed, we can use the Poisson distribution formula: \[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \] where: - \( P(X = k) \) is the probability of \( k \) events occurring in a given time period, - \( \lambda \) is the average number of events that occur in a given time period, - \( k \) is the number of events we are interested in, - \( e \) is the base of the natural logarithm. Given that the mean number of radioactive atoms that decayed in a day is 56.775, we can use this value as \( \lambda \) in the formula. Let's calculate the probability that on a given day, 53 radioactive atoms decayed. Simplify the expression by following steps: - step0: Solution: \(\frac{\left(\frac{1}{e}\times 56.775^{53}\right)}{53!}\) - step1: Remove the parentheses: \(\frac{\frac{1}{e}\times 56.775^{53}}{53!}\) - step2: Convert the expressions: \(\frac{\frac{1}{e}\times \left(\frac{2271}{40}\right)^{53}}{53!}\) - step3: Multiply the numbers: \(\frac{\frac{2271^{53}}{40^{53}e}}{53!}\) - step4: Multiply by the reciprocal: \(\frac{2271^{53}}{40^{53}e}\times \frac{1}{53!}\) - step5: Multiply the terms: \(\frac{2271^{53}}{40^{53}e\times 53!}\) The probability that on a given day, 53 radioactive atoms decayed is approximately \( \frac{2271^{53}}{40^{53}e \times 53!} \). To round this to six decimal places, we can calculate the value of this expression. Simplify the expression by following steps: - step0: Solution: \(\frac{2271^{53}}{\left(40^{53}e\times 53!\right)}\) - step1: Remove the parentheses: \(\frac{2271^{53}}{40^{53}e\times 53!}\) The probability that on a given day, 53 radioactive atoms decayed is approximately 0.000000.