Discrete Probability Distributions

Main idea

A random variable turns each outcome of a chance experiment into a number, like the number of heads in three flips. When it can only take separate values you can list, you describe it with a table of each value and its probability, and those probabilities have to add up to 1.

0 \le {\htmlData{sym=0}{P(x)}} \le 1, \qquad \sum {\htmlData{sym=0}{P(x)}} = 1

What the symbols mean

$P(x)$
Probability of the value x
How likely the random variable is to equal x.

Explanation

A random variable is simply a number whose value depends on chance, like the number of heads when you flip three coins. It's "discrete" when its possible values are separate and countable (0, 1, 2, 3 heads, never 1.7), and a probability distribution is just the list of those values with how likely each one is. The two rules look formal, but they're common sense: $0 \le P(x) \le 1$ (no probability is negative or more than certain) and $\sum P(x) = 1$ (something on the list has to happen). That's all a valid distribution needs.

Notation: capital $X$ is the random variable, lowercase $x$ a particular value, and $P(X = x)$ or $P(x)$ its probability.

Example: $X$ = number of heads in 3 coin flips. There are 8 equally likely sequences (HHH, HHT, ...), and you get the table by counting how many sequences give each number of heads.

$x$	0	1	2	3
$P(x)$	$\frac18$	$\frac38$	$\frac38$	$\frac18$

Probabilities of ranges add: $P(X \ge 2) = \frac38 + \frac18 = \frac12$ . That works because "2 heads" and "3 heads" can't both happen, the addition rule for disjoint events from the probability unit.

A valid distribution has no negative probabilities and totals exactly 1.

A common mistake. For "at least 2," it's tempting to add only the probability of exactly 2. "At least 2" means 2 or more, so you add every value from 2 up. Likewise "fewer than 1" means only 0.

See it

This is the distribution of the sum of two dice. It peaks at 7 because 7 has the most ways to happen, 6 of the 36 rolls, and falls off evenly to 2 and 12, which have one way each. The shaded bars are the event 'sum at least k', so its probability is their total height: 21/36 for k = 7, and just 6/36 by k = 10.

Worked examples

Example 1

$X$ takes the values 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4 and $p$ . Find $p$ and $P(X \ge 2)$ .

1
The probabilities have to add to 1: $0.1 + 0.3 + 0.4 + p = 1$ , so $p = 0.2$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

$X$ takes values 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, $p$ . Find $p$ .

Problem 2+5 XP

For the same $X$ , find $P(X \ge 2)$ .

Problem 3+5 XP

For the same $X$ , find $P(X < 1)$ .

Problem 4+5 XP

For 3 coin flips, find $P(\text{exactly 2 heads})$ .

Problem 5+5 XP

Is $P(0) = 0.5$ , $P(1) = 0.4$ , $P(2) = 0.2$ a valid distribution?

Problem 6+5 XP

A food truck sells 0, 1, 2 or 3 specials in an hour with probabilities 0.2, 0.4, 0.3 and 0.1. What's the probability it sells at least 2 in an hour?

Problem 7+5 XP

Roll two fair dice. What's the probability the sum is at least 10?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Roll three fair dice. What's the probability the sum is exactly 10?

Where it's used

The number of customers in line, defective items in a batch and goals in a match are discrete random variables, and businesses plan around their distributions.

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