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

Find the probability that a student participates in exactly 2 activities.

Answer 1

To find the probability that a student participates in exactly 2 activities, we look at the probability distribution provided for the random variable \(X\).

According to the distribution:

\[ P(X = 2) = 0.47 \]

Therefore, the probability that a student participates in exactly 2 activities is 0.47.

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 participating in 0, 1, and 2 activities.

From the given probability distribution:

\[ P(X = 0) = 0.04 \] \[ P(X = 1) = 0.14 \] \[ P(X = 2) = 0.47 \]

Now, we add these probabilities together:

\[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \] \[ P(X < 3) = 0.04 + 0.14 + 0.47 = 0.65 \]

Therefore, the probability that a student participates in less than three activities is 0.65.

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 sum the probabilities of participating in 2, 3, and 4 activities.

From the given probability distribution:

\[ P(X = 2) = 0.47 \] \[ P(X = 3) = 0.22 \] \[ P(X = 4) = 0.13 \]

Now, we add these probabilities together:

\[ P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) \] \[ P(X \geq 2) = 0.47 + 0.22 + 0.13 = 0.82 \]

Therefore, the probability that a student participates in at least two activities is 0.82.