Question

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