Question

imagine selecting a u.s. high school student at random. define the random variable x = number of languages spoken by the student. the table gives the probability distribution of x.
languages | 1 | 2 | 3 | 4 | 5
probability | 0.630 | 0.295 | 0.065 | 0.008 | 0.002

(a) is x a discrete or a continuous random variable? explain your answer.
\\(\bigcirc\\) x is a continuous random variable because it can take any value in an interval on the number line.
\\(\bigcirc\\) x is a discrete random variable because it takes a fixed set of numerical values with gaps between them.
\\(\bigcirc\\) x is a discrete random variable because it can take any value in an interval on the number line.
\\(\bigcirc\\) x is a continuous random variable because it takes a fixed set of numerical values with gaps between them.

(b) find \\(p(x \geq 3)\\). explain in words what this probability means.
\\(\bigcirc\\) \\(p(x \geq 3) = 0.010\\). this is the probability that a randomly selected u.s. high school student speaks more than 3 languages.
\\(\bigcirc\\) \\(p(x \geq 3) = 0.075\\). this is the probability that a randomly selected u.s. high school student speaks at least 3 languages.
\\(\bigcirc\\) \\(p(x \geq 3) = 0.075\\). this is the probability that a randomly selected u.s. high school student speaks more than 3 languages.
\\(\bigcirc\\) \\(p(x \geq 3) = 0.010\\). this is the probability that a randomly selected u.s. high school student speaks at least 3 languages.

(c) find the probability that a randomly selected student speaks fewer than 3 languages.
(include 3 decimal places.)

Explanation:

Part (a)

Step1: Recall definitions

Discrete random variables take distinct, separate values (like counts: 1, 2, 3,...) with gaps. Continuous variables take any value in an interval (e.g., height, weight).

Step2: Analyze \( X \)

\( X \) is the number of languages spoken, so values are 1, 2, 3, 4, 5 (fixed set, gaps between them). So \( X \) is discrete.

Step1: Define \( P(X \geq 3) \)

\( P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5) \).

Step2: Substitute probabilities

From the table: \( P(3) = 0.065 \), \( P(4) = 0.008 \), \( P(5) = 0.002 \).
Sum: \( 0.065 + 0.008 + 0.002 = 0.075 \).
"At least 3" means 3 or more.

Step1: Define "fewer than 3"

"Fewer than 3" means \( X = 1 \) or \( X = 2 \).

Step2: Sum probabilities

\( P(X < 3) = P(X = 1) + P(X = 2) \).
From the table: \( P(1) = 0.630 \), \( P(2) = 0.295 \).
Sum: \( 0.630 + 0.295 = 0.925 \).

Answer:

\( X \) is a discrete random variable because it takes a fixed set of numerical values with gaps between them. (The second option)

Part (b)