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:

$$P(X\ge 2)=P(X=2)+P(X=3)+P(X=4)$$ $$P(X\ge 2)=0.47+0.22+0.13=0.82$$

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