Compute Probabilities of Binomial Experiments
-We believe that 87% of the population of all Calculus I students consider calculus an exciting subject. Suppose we randomly and independently selected 25 students from the population. If the true percentage is really 87%, find the probability of observing 24 or more of the students who consider calculus to be an exciting subject in our sample of 25.
A) 0.145666
B) 0.030760
C) 0.114907
D) 0.854334

Question

Quizplus · Accepted Answer

The probability of observing 24 or more students who consider calculus exciting in a sample of 25, given a success rate of 87%, can be calculated using the binomial probability formula or a binomial distribution table. However, for this specific question, it's more efficient to calculate the probability of exactly 24 students and exactly 25 students finding calculus exciting and then summing those probabilities. Given the binomial formula $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, where $n=25$, $p=0.87$, and $k$ is either 24 or 25, we calculate $P(X=24)$ and $P(X=25)$ and sum them. The exact calculations might be complex, but the correct choice is based on the options provided, which indicates that the sum of these probabilities is closest to the value in option A.