Questions: Next The following table represents undergraduate students at a local university. One student is chosen at random. Freshman Sophomore Junior Senior Total ----------------------------------------------------------- Live On Campus 3106 2539 3018 1432 10095 Live Off Campus 3102 3418 2257 1627 10404 Total 6208 5957 5275 3059 20499 Answer the following using either fractions or decimals rounded to three places a) Find the probability that the student lives off campus or is a junior. b) Find the probability that the student lives on campus and is a senior. c) Find the probability that the student is a freshman given that they live on campus.

Solution Steps

To solve the given probability questions, we will use the data from the table and apply basic probability formulas.

a) To find the probability that the student lives off campus or is a junior, we will use the formula for the union of two events: P(A or B) = P(A) + P(B) - P(A and B).

b) To find the probability that the student lives on campus and is a senior, we will use the formula for the intersection of two events: P(A and B) = Number of favorable outcomes / Total number of outcomes.

c) To find the probability that the student is a freshman given that they live on campus, we will use the conditional probability formula: P(A|B) = P(A and B) / P(B).

To find the probability that the student lives off campus or is a junior, we use the formula for the union of two events: \[ P(\text{Off Campus or Junior}) = P(\text{Off Campus}) + P(\text{Junior}) - P(\text{Off Campus and Junior}) \] Given: \[ P(\text{Off Campus}) = \frac{10404}{20499} \approx 0.5075 \] \[ P(\text{Junior}) = \frac{5275}{20499} \approx 0.2573 \] \[ P(\text{Off Campus and Junior}) = \frac{2257}{20499} \approx 0.1101 \] Thus: \[ P(\text{Off Campus or Junior}) \approx 0.5075 + 0.2573 - 0.1101 = 0.6548 \]

To find the probability that the student lives on campus and is a senior, we use the formula for the intersection of two events: \[ P(\text{On Campus and Senior}) = \frac{1432}{20499} \approx 0.0699 \]

To find the probability that the student is a freshman given that they live on campus, we use the conditional probability formula: \[ P(\text{Freshman | On Campus}) = \frac{P(\text{Freshman and On Campus})}{P(\text{On Campus})} \] Given: \[ P(\text{Freshman and On Campus}) = \frac{3106}{10095} \approx 0.3077 \]

Final Answer