Following is the probability distribution of a random variable that represents the number of extracurricular activities a college freshman participates in.

Question

x| 0 1 2 3 4
P(x)| 0.05 0.14 0.45 0.23 0.13

(a) Find the probability that a student participates in exactly three activities.

Answer 1

To find the probability that a student participates in exactly three activities, we can refer directly to the given probability distribution.

From the provided information, we can see:

\[ P(3) = 0.23 \]

Thus, the probability that a student participates in exactly three activities is 0.23.

Answer 2

(b) Find the probability that a student participates in less than three activities.

Answer 3

To find the probability that a student participates in less than three activities, we need to sum the probabilities of the student participating in 0, 1, or 2 activities.

From the probability distribution:

• \( P(0) = 0.05 \)
• \( P(1) = 0.14 \)
• \( P(2) = 0.45 \)

Now, we sum these probabilities:

\[ P(X < 3) = P(0) + P(1) + P(2) \]

Calculating this gives:

\[ P(X < 3) = 0.05 + 0.14 + 0.45 = 0.64 \]

Therefore, the probability that a student participates in less than three activities is 0.64.

Answer 4

(c) Find the probability that a student participates in at least two activities.

Answer 5

To find the probability that a student participates in at least two activities, we need to consider the probabilities of participating in 2, 3, or 4 activities.

From the probability distribution, we have:

• \( P(2) = 0.45 \)
• \( P(3) = 0.23 \)
• \( P(4) = 0.13 \)

Now, we sum these probabilities:

\[ P(X \geq 2) = P(2) + P(3) + P(4) \]

Calculating this gives:

\[ P(X \geq 2) = 0.45 + 0.23 + 0.13 = 0.81 \]

Therefore, the probability that a student participates in at least two activities is 0.81.