To determine whether the random variable [tex]\( x \)[/tex] is discrete or continuous, we need to understand the nature of the variable and the context provided in the question.
In this context, [tex]\( x \)[/tex] represents the total number of die rolls required for an individual to roll a five.
Here are the steps to make the determination:
1. Understand Possible Values of [tex]\( x \)[/tex]:
- [tex]\( x \)[/tex] accounts for each roll of the die until a five is rolled.
- The number of rolls can be 1, 2, 3, and so on.
2. Definition of Discrete and Continuous Variables:
- A discrete random variable is one that can only take on a finite or countably infinite set of values. Typically, these are values that one can list: such as 1, 2, 3, etc.
- A continuous random variable is one that can take any value within a given range. This range can include any possible value like 0.1, 2.35, 7.89, etc., where fractions and decimals are involved and there are no gaps between possible values.
3. Application to [tex]\( x \)[/tex]:
- Since you can count the number of die rolls, and they can only take on whole number values (you can’t have 3.5 rolls or 2.7 rolls), [tex]\( x \)[/tex] can only be 1, 2, 3, ... ad infinitum.
- There are no intermediate values possible between the die rolls; you either make the roll, or you don’t. Therefore, there are distinct, separate values without fractions or decimals involved.
Based on the above steps, we conclude that the random variable [tex]\( x \)[/tex] cannot take on fractional values and only takes integer counts of die rolls.
Thus, the random variable [tex]\( x \)[/tex] is Discrete.
