Which of the following is/are discrete random variables?

B

A student's height and the speed of a car can take values over a continuum and hence, are not discrete variables.

To determine which of the given variables are discrete random variables, let's define what a discrete random variable is. A discrete random variable is a variable that can only take on specific values from a countable set, typically represented by integers. Discrete random variables have gaps or jumps between possible values.

Let's analyze each option one by one:

I. The height of a student selected at random from 217 students. This variable represents a continuous measurement, as height can take on any value within a range. Therefore, it is not a discrete random variable.

II. The slot number where the ball comes to rest in a roulette wheel. In a roulette wheel, there are a finite number of slots, typically numbered from 0 to 36 (in European roulette) or 0 to 37 (in American roulette, which includes an additional slot for the double zero). These slot numbers are discrete values, and the ball can only land in one of these specific slots. Hence, this variable is a discrete random variable.

III. The color of a playing card drawn at random from a pack. A standard deck of playing cards has 52 cards, divided into four suits (hearts, diamonds, clubs, and spades), with each suit having 13 cards (Ace, 2-10, Jack, Queen, and King). The color of a playing card can be red (hearts and diamonds) or black (clubs and spades). The variable represents a finite set of values and is discrete. Therefore, this variable is a discrete random variable.

IV. The speed of a car tracked by the radar gun of highway patrol. The speed of a car can take on any value within a range, making it a continuous variable. As such, it is not a discrete random variable.

Based on the analysis above, the discrete random variables among the given options are:

• II. The slot number where the ball comes to rest in a roulette wheel.
• III. The color of a playing card drawn at random from a pack.