Probability distributions

Discrete random variables, probability distribution tables and graphs, the condition that probabilities add to 1, and relative frequencies as estimates of probabilities.

The language

A discrete random variable $X$ takes separate values, usually counts, each with a probability. Its probability distribution lists every value with its probability. Two conditions always hold: $$0 \le P(X = x) \le 1 \quad \text{and} \quad \sum P(X = x) = 1.$$

Example 1: two coins●

Two fair coins are tossed and $X$ is the number of heads. Write the distribution.

The outcomes HH, HT, TH, TT are equally likely.

$x$	0	1	2
$P(X = x)$	$\tfrac14$	$\tfrac12$	$\tfrac14$

Example 2: an unknown probability●●

Find $k$ and $P(X \ge 3)$ for this distribution.

$x$	1	2	3	4
$P(X = x)$	$0.1$	$0.3$	$k$	$0.2$

$0.1 + 0.3 + k + 0.2 = 1$, so $k = 0.4$.

$P(X \ge 3) = 0.4 + 0.2 = 0.6$.

Relative frequency

When an experiment is repeated many times, the relative frequency of a value is $\dfrac{\text{frequency}}{\text{number of trials}}$. It estimates the probability, and the estimate usually improves as the number of trials grows.

Example 3●

A spinner lands on red $46$ times in $200$ spins. Estimate $P(\text{red})$.

$\dfrac{46}{200} = 0.23$

Flashcards

Conditions for a distribution

Every probability is between $0$ and $1$, and they add to $1$.

Relative frequency

$\dfrac{\text{frequency}}{\text{number of trials}}$

Discrete random variable

Takes separate values, usually counts.

Practice

In Example 2, find $k$.
In Example 2, find $P(X \ge 3)$.
In Example 2, find $P(X \lt 3)$.
Two fair coins are tossed. Find $P(\text{exactly one head})$.
A spinner lands on red $46$ times in $200$ spins. Estimate $P(\text{red})$.
Which list could be a probability distribution?
Which is a discrete random variable?
$X$ is the score on a fair die. Find $P(X = 4)$.
As the number of trials increases, the relative frequency: