Problem

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?

Solution

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 :\(\boxed{\text{The probability that more than 40% of the 55 college students will change their major is approximately 0.6206}}\)

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Source: https://solvelyapp.com/problems/30616/

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