Discrete Uniform Distribution

Equal Probability for Every Outcome

On one toss of a fair coin, heads and tails have equal probability. Each receives half of the total probability. A discrete uniform distribution extends this idea to a finite collection of outcomes.

A probability distribution pairs each value with its probability of occurring. In a sample space $S$ whose outcomes are all equally likely, an event $A$ has probability

P(A)=\frac{n(A)}{n(S)}

Here $n(A)$ counts outcomes belonging to the event, while $n(S)$ counts every possible outcome. The condition all outcomes are equally likely must hold before you use this ratio.

Probability of One Value

Suppose a random variable $X$ has $k$ equally likely values in a set $S$ . Its probability mass function assigns the same share to each value:

P(X=x)=\frac{1}{k},\qquad x\in S

Symbol	Meaning	Example with a fair die
$x$	One possible value	The number $3$
$k$	Number of possible values	Six faces
$P(X=x)$	Probability of that value	$P(X=3)=1/6$

The probabilities must sum to one. There are $k$ shares of size $1/k$ , giving $k(1/k)=1$ . This model uses a finite set of values. A continuous uniform distribution uses a density over an interval, so probability comes from area rather than counting values.

Comparing a Die and a Spinner

In a uniform model, increasing the number of possible outcomes reduces the probability of one outcome. A die divides the total probability into six shares, while a wheel with eight sectors divides it into eight shares.

One Roll of a Die

A fair die has sample space $S=\{1,2,3,4,5,6\}$ . For every $x\in S$ , we obtain

P(X=x)=\frac{1}{6}\approx16.67\%

Rolling $1$ is as likely as rolling $6$ . However, the event of rolling an even number contains three outcomes: $2,4,6$ . Thus one value and one event can have different probabilities:

P(X\text{ even})=\frac{3}{6}=\frac{1}{2}

A Wheel with Eight Sectors

A fair spinner has eight equal sectors numbered $1$ through $8$ . Compare sector $5$ with the others. Every sector has central angle $360^\circ/8=45^\circ$ .

Eight equal sectors

Each number occupies one 45-degree sector. All sectors have equal area.

The sample space is $S=\{1,2,3,4,5,6,7,8\}$ , so $k=8$ . Each sector is equally likely, giving the mass function $P(X=x)=1/8$ for every $x\in S$ . For sector five,

P(X=5)=\frac{1}{8}=0.125=12.5\%

Exercises

Identify the outcome set, count the equally likely outcomes, and find the requested event probability. Check that the requested object belongs to the outcome set before dividing.

A bag contains ten balls identical except for color. There is one red, one blue, one green, one yellow, and six more balls each with a different color. Each ball is equally likely to be drawn. Write the probability mass function for the ball color and calculate the probability of yellow.
A standard bridge deck contains 52 distinct cards, each equally likely to be drawn. Identify the probability distribution for drawing one card and calculate the probability of the King of Hearts.

Yellow Ball

There are ten colors, so $k=10$ . For each color $x$ in the color set $S$ ,

P(X=x)=\frac{1}{10},\qquad x\in S

Drawing yellow corresponds to exactly one outcome. Therefore,

P(\text{yellow ball})=\frac{1}{10}=0.1=10\%

King of Hearts

Drawing one card follows a discrete uniform distribution because all 52 cards are equally likely. Only one of those 52 outcomes is the King of Hearts.

P(\text{King of Hearts})=\frac{1}{52}\approx0.01923\approx1.92\%