Example-4: It has been estimated that $43 \%$ of all college students change their major at least once during the course of their college career. If we take a random sample of 55 college students, what is the probability that more than $40 \%$ will change their major?
Answer
\(\boxed{\text{The probability that more than 40% of the 55 college students will change their major is approximately 0.6206}}\)
Step 1 ：Let n be the number of students, p be the probability of changing major, and threshold be the number of students changing major for more than 40%.
Step 2 ：n = 55
Step 3 ：p = 0.43
Step 4 ：threshold = 0.4 * 55 = 22
Step 5 ：Use the binomial distribution to find the probability of more than 22 students changing their major.
Step 6 ：prob = 1 - P(X \leq 22) = 1 - \sum_{k=0}^{22} \binom{55}{k} (0.43)^k (1-0.43)^{55-k}
Step 7 ：prob \approx 0.6206
Step 8 ：\(\boxed{\text{The probability that more than 40% of the 55 college students will change their major is approximately 0.6206}}\)
