Final Answer:
a. The probability that exactly 26 students need to take another math class is approximately 0.1683.
b. The probability that at most 26 students need to take another math class is approximately 0.2499.
c. The probability that at least 22 students need to take another math class is approximately 0.8025.
d. The probability that between 18 and 24 students (including 18 and 24) need to take another math class is approximately 0.0597.
Step-by-step explanation:
a. To find the probability of exactly 26 students needing another math class, we use the binomial probability formula: P(X = k) = C(n, k) p^k q^(n-k). Here, n is the number of trials (36 students), k is the number of successes (26 students), p is the probability of success (68% or 0.68), and q is the probability of failure (1 - p). Substituting these values and calculating gives us the answer.
b. The probability of at most 26 students needing another math class is the sum of the probabilities for 0 to 26 students. We can find this by summing the individual probabilities using the formula mentioned above.
c. The probability of at least 22 students needing another math class is the complement of the probability that fewer than 22 students need another math class. So, P(X ≥ 22) = 1 - P(X < 22), which is calculated using the binomial probability formula.
d. To find the probability of between 18 and 24 students (inclusive) needing another math class, we sum the individual probabilities for 18 to 24 using the same binomial probability formula.