To find the probability that a student scores higher than 85 on the exam, we need to calculate the area under the normal distribution curve to the right of the mean (85).
We can use the standard normal distribution and the Z-score formula to find this probability. The Z-score is calculated as (X - μ) / σ, where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation. In this case, the Z-score is (85 - 85) / 10 = 0. The probability can be found by looking up the Z-score in the standard normal distribution table or using a calculator, which gives a probability of 0.5. Therefore, the probability that a student scores higher than 85 on the exam is 0.5.
Assuming exam scores are independent and 10 students take the exam, we can consider each student's score as a separate event. The probability that a student scores 85 or higher can be found using the Z-score formula as described in part a. For each student, the Z-score would be (85 - 85) / 10 = 0, and the probability of scoring 85 or higher is 0.5.
Since the students' scores are independent, the probability that 4 or more students score 85 or higher can be calculated using the binomial distribution formula. Using the binomial probability formula with n = 10 (number of students), p = 0.5 (probability of scoring 85 or higher for each student), and k = 4 (or more), we can calculate the probability.