Question

An experienced teacher writes an exam so that, on average, about 5% of students will earn an A grade. If she has 50 students in her class and their performance is independent, what is the probability that at least one student gets an A?

Answer 1

To solve this problem, we can use the complement rule in probability. The steps are as follows:

1. **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:
\[
q = 1 - p = 1 - 0.05 = 0.95.
\]

2. **Determine the Number of Students**: The teacher has a total of \( n = 50 \) students in her class.

3. **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:
\[
P(\text{no A's}) = q^n = (0.95)^{50}.
\]

4. **Calculate the Probability**:
We need to calculate \( (0.95)^{50} \):
\[
(0.95)^{50} \approx 0.2146 \quad (\text{using a calculator}).
\]

5. **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.