Is "The number of points scored during a basketball game." discrete or continuous?

A random variable is considered discrete if its possible values are countable. In a basketball game, for example, it is only possible for a team's score to be a whole number—no fractions or decimals are allowed, and so the score is discrete.

In statistics, it is possible for a discrete random variable to take on a specific, single value with non-zero probability. For example, if we let #S# be a basketball team's score at the end of a game, it is possible for #S# to be 75, thus #P(S = 75) > 0#.

On the other hand, a continuous random variable is one where the range of its possible values is uncountable. Good examples are time, length, mass, temperature, etc. It is not possible for an event to occur exactly at 4:27pm, or a person's height #H# to be exactly 1.65 meters. There will always be some decimal places that we can't measure. In statistics, this is written as #P(H=1.65)=0#.

For a continuous random variable, we must talk about the probability of it being within a range of values. We could find a value for #P(H>1.5)#, or perhaps #P(1.3 < H < 1.6)#. But, just like the area of a line segment is 0 because it has no width, the probability of #H# being an exact value is zero as well.

A good rule of thumb is this: if the variable you're measuring has to be rounded before it's written down, then it's continuous. If no rounding is necessary, as with anything that's countable, then it's discrete.