Question

There are undergraduates registered at a certain college. Of them, are taking one course, are taking two courses, are taking three courses, are taking four courses, are taking five courses, and are taking six courses. Let be the number of courses taken by a student randomly sampled from this population. Find the probability distribution of . Round the answers to four decimal places as needed.

Answer 1

To find the probability distribution for the number of courses taken by a randomly sampled student, we need to calculate the probabilities for each possible value of X (the number of courses taken). Let's denote P(X = x) as the probability that a student is taking x courses.

You mentioned the following counts:

300 students are taking one course (X = 1).

200 students are taking two courses (X = 2).

150 students are taking three courses (X = 3).

100 students are taking four courses (X = 4).

50 students are taking five courses (X = 5).

20 students are taking six courses (X = 6).

To calculate the probability of a student taking a specific number of courses (P(X = x)), divide the number of students taking that number of courses by the total number of undergraduates.

The total number of undergraduates is:

Total = 300 + 200 + 150 + 100 + 50 + 20 = 820

Now, calculate the probabilities:

P(X = 1) = 300 / 820

P(X = 2) = 200 / 820

P(X = 3) = 150 / 820

P(X = 4) = 100 / 820

P(X = 5) = 50 / 820

P(X = 6) = 20 / 820

Now, calculate these probabilities:

P(X = 1) = 0.3659 (rounded to four decimal places)

P(X = 2) = 0.2439 (rounded to four decimal places)

P(X = 3) = 0.1829 (rounded to four decimal places)

P(X = 4) = 0.1219 (rounded to four decimal places)

P(X = 5) = 0.0609 (rounded to four decimal places)

P(X = 6) = 0.0244 (rounded to four decimal places)

So, the probability distribution of the number of courses taken by a randomly sampled student is as follows:

P(X = 1) ≈ 0.3659

P(X = 2) ≈ 0.2439

P(X = 3) ≈ 0.1829

P(X = 4) ≈ 0.1219

P(X = 5) ≈ 0.0609

P(X = 6) ≈ 0.0244

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