Probability distribution

A probability distribution says how the total probability of 1 is divided among the values a random variable can take. For a discrete variable it is a table or a formula giving $P(X = x)$ for every possible $x$, and two conditions define it: each probability satisfies $0 \le P(X = x) \le 1$, and $\sum P(X = x) = 1$. For a continuous variable there is no table. Probability comes from area under a density curve, and the area under the whole curve is 1.

Let $X$ be the number of people in a car passing a checkpoint, with $P(1) = 0.35$, $P(2) = 0.30$, $P(3) = 0.20$, and the last probability left blank. The sum condition fills it in: $P(4) = 1 - (0.35 + 0.30 + 0.20) = 0.15$. That is not a trick, it is the definition doing the work, and it is also why a table whose entries add to 0.97 is not a probability distribution at all.

The misreading to watch is "I rolled the die 60 times and got twelve 3s, so $P(X = 3) = 0.20$." That is a relative frequency from one sample, not the distribution. A probability distribution is what the chance process does over the long run and it is fixed before any data exist; a sample wobbles around it and settles closer as the number of trials grows.

The discrete and continuous cases part company over a single value. For the car table, $P(X \le 2) = 0.65$ while $P(X < 2) = 0.35$, because $X = 2$ carries real probability of its own. For a continuous variable, $P(X = x) = 0$ at every single $x$, so $P(X \le 2)$ and $P(X < 2)$ are equal. Reading a strict inequality as if it were the other kind is a genuine error for counts and harmless for measurements.

Probability distributions enter the course at topic 2.8, Introduction to Random Variables and Probability Distributions.