Final answer:
To find the probabilities in each case, we can use the binomial probability formula. By plugging in the values of n, x, and p, we can calculate the desired probabilities. For part a, the probability is approximately 0.00008. For parts b, c, and d, we need to sum up individual probabilities using the binomial probability formula.
Step-by-step explanation:
To find the probability in each case, we can use the binomial probability formula:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
where n is the number of trials, x is the number of successes, p is the probability of success in one trial, and C(n, x) is the number of combinations.
a. In this case, n = 38, x = 29, and p = 0.72. Plugging these values into the formula, we get:
P(29) = C(38, 29) * 0.72^29 * (1-0.72)^(38-29)
= (38!)/(29! * (38-29)!) * 0.72^29 * 0.28^9
Using a calculator, we can evaluate this probability to be approximately 0.00008.
b. To find the probability of fewer than 27 students needing to take another math class, we need to calculate the probabilities of 0 to 26 students needing another math class and sum them up:
P(<27) = P(0) + P(1) + P(2) + ... + P(26)
We can use the binomial probability formula to calculate each individual probability and then add them up using a calculator.
c. To find the probability of more than 25 students needing to take another math class, we need to calculate the probabilities of 26 to 38 students needing another math class and sum them up:
P(>25) = P(26) + P(27) + P(28) + ... + P(38)
Again, we can use the binomial probability formula to calculate each individual probability and then add them up using a calculator.
d. To find the probability of between 26 and 30 (including 26 and 30) students needing to take another math class, we need to calculate the probabilities of 26 to 30 students needing another math class and sum them up:
P(26-30) = P(26) + P(27) + P(28) + P(29) + P(30)
Once again, we can use the binomial probability formula to calculate each individual probability and then add them up using a calculator.