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)| 0.04 0.14 0.47 0.22 0.13
(a) Find the probability that a student participates in exactly three activities.

The probability that a student participates in exactly three activities is

Answer 1

To find the probability that a student participates in exactly three activities, we can look at the provided probability distribution.

From the table given:

• For \( x = 3 \), \( P(x) = 0.22 \).

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

\[ \text{The probability that a student participates in exactly three activities is } 0.22. \]

Answer 2

b) Find the probability that a student participates in more than two activities.

Answer 3

To find the probability that a student participates in more than two activities, we need to consider the probabilities for \( x = 3 \) and \( x = 4 \).

From the given probability distribution:

• \( P(3) = 0.22 \) (for 3 activities)
• \( P(4) = 0.13 \) (for 4 activities)

Now, we can sum these probabilities:

\[ P(x > 2) = P(3) + P(4) = 0.22 + 0.13 = 0.35 \]

Therefore, the probability that a student participates in more than two activities is:

\[ \text{The probability that a student participates in more than two activities is } 0.35. \]

Answer 4

(c) Find the probability that a student participates in at least one activity.

Answer 5

To find the probability that a student participates in at least one activity, we need to consider the probabilities for \( x = 1, 2, 3, \) and \( 4 \). Alternatively, since we know the total probability must sum to 1, we can also find this probability by subtracting the probability of not participating in any activities (i.e., \( x = 0 \)) from 1.

From the given probability distribution:

• \( P(0) = 0.04 \) (for 0 activities)

Now, we can calculate the probability of participating in at least one activity:

\[ P(x \geq 1) = 1 - P(0) = 1 - 0.04 = 0.96 \]

Therefore, the probability that a student participates in at least one activity is:

\[ \text{The probability that a student participates in at least one activity is } 0.96. \]