To solve this problem, we can use the complement rule in probability. The steps are as follows:
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Define the Probability of an Individual Student Earning an A: We know that the probability (p) of a student earning an A is 5%, or \( p = 0.05 \). Thus, the probability of a student not earning an A is: \[ q = 1 - p = 1 - 0.05 = 0.95. \]
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Determine the Number of Students: The teacher has a total of \( n = 50 \) students in her class.
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Calculate the Probability that No Students Earn an A: The probability that none of the 50 students earn an A (all students not earning an A) is given by the product of the probabilities of each student not earning an A: \[ P(\text{no A's}) = q^n = (0.95)^{50}. \]
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Calculate the Probability: We need to calculate \( (0.95)^{50} \): \[ (0.95)^{50} \approx 0.2146 \quad (\text{using a calculator}). \]
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Find the Probability that at Least One Student Earns an A: We now use the complement of the probability that no students earn an A: \[ P(\text{at least one A}) = 1 - P(\text{no A's}) = 1 - (0.95)^{50}. \] Calculating: \[ P(\text{at least one A}) \approx 1 - 0.2146 \approx 0.7854. \]
Thus, the final answer is: \[ \boxed{0.7854} \]
This means there is approximately a 78.54% probability that at least one student in the class will achieve an A grade.
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