The applicants to a university were surveyed about their planned living arrangements for the coming year. The results of the survey are in the frequency table below.
| | On-Campus | Off-Campus | Total |
| :-------------------- | :-------- | :--------- | :---- |
| Transfer Applicants | 38 | 66 | 104 |
| Freshman Applicants | 85 | 52 | 137 |
| Total | 123 | 118 | 241 |
What is the probability that an applicant planning to stay off-campus is a transfer applicant?
A. 0.525
B. 0.635
C. 0.490
D. 0.559
Answer (1)
Define events: T (transfer applicant), O (off-campus).
Use conditional probability formula: P ( T ∣ O ) = P ( O ) P ( TO ) .
Calculate probabilities: P ( TO ) = 241 66 , P ( O ) = 241 118 .
Find conditional probability: P ( T ∣ O ) = 118 66 ≈ 0.559 .
Explanation
Analyze the problem and data We are given a frequency table that summarizes the living arrangements of university applicants. The table categorizes applicants as either transfer or freshman and indicates whether they plan to live on-campus or off-campus. We are asked to find the probability that an applicant is a transfer applicant, given that they plan to stay off-campus.
Define events and conditional probability formula Let T be the event that an applicant is a transfer applicant, and let O be the event that an applicant plans to stay off-campus. We want to find the conditional probability P ( T ∣ O ) , which is the probability that an applicant is a transfer applicant given that they plan to stay off-campus. The formula for conditional probability is: P ( T ∣ O ) = P ( O ) P ( TO )
Determine the probability of T and O From the table, we can see that there are 66 applicants who are transfer applicants and plan to stay off-campus. The total number of applicants surveyed is 241. Therefore, the probability of an applicant being a transfer applicant and planning to stay off-campus is: P ( TO ) = 241 66
Determine the probability of O From the table, we can see that the total number of applicants who plan to stay off-campus is 118. Therefore, the probability of an applicant planning to stay off-campus is: P ( O ) = 241 118
Calculate the conditional probability Now, we can substitute these values into the conditional probability formula: P ( T ∣ O ) = 241 118 241 66 = 118 66 To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2: P ( T ∣ O ) = 118 ÷ 2 66 ÷ 2 = 59 33 Now, we can convert this fraction to a decimal to compare with the given options: 59 33 ≈ 0.559322
State the final answer Comparing the calculated probability to the given options, we find that the closest option is 0.559. Therefore, the probability that an applicant planning to stay off-campus is a transfer applicant is approximately 0.559.
Examples
Conditional probability is used in various real-life scenarios, such as in medical diagnosis to determine the probability of a disease given certain symptoms, in finance to assess the risk of an investment given market conditions, and in marketing to predict customer behavior based on past actions. In this case, the university might use this probability to understand the needs and preferences of transfer students living off-campus to better support them.
