Discrete Uniform Distribution

Equal Probability for Every Outcome

Many everyday situations have uncertain outcomes. A fair coin, for example, can land heads or tails, and the two outcomes are equally likely.

When a finite sample space contains equally likely outcomes, the probability of an event $A$ is the number of outcomes in $A$ divided by the total number of outcomes in the sample space $S$.

Modeling a Finite Set of Outcomes

In a discrete uniform distribution, every possible outcome has exactly the same probability. The discussion below focuses on a discrete model with a finite outcome set. A continuous uniform distribution uses a density over an interval.

For a fair six-sided die, the possible outcomes are $1, 2, 3, 4, 5, 6$. Rolling a $1$ is just as likely as rolling a $6$. Each outcome therefore has probability $\frac{1}{6}$.

Mathematical Formula

Suppose a random variable $X$ has a finite set $S$ of $k$ possible values and all of them are equally likely. Then its probability mass function is:

Where:

• $P(X=x)$ is the probability that $X$ takes the particular value $x$

• $x$ is one of the possible outcomes (for example, the number $3$ on a die)

• $k$ is the total number of possible outcomes (for example, $6$ for a die)

Each outcome receives an equal share of the total probability: one divided by the number of possible outcomes.

Equal Probabilities in Finite Sample Spaces

The following examples apply the definition to finite sample spaces.

Rolling a Die

For example, we have a balanced six-sided die that is rolled once. How do we determine its uniform distribution?

• Possible Outcomes: The sample space is $\{1, 2, 3, 4, 5, 6\}$

• Total Number of Outcomes: There are $6$ possible outcomes, so $k=6$

• Probability Mass Function:

  $$P(X=x) = \frac{1}{6}, \qquad x \in \{1,2,3,4,5,6\}$$

• Meaning: The probability of getting the numbers $1, 2, 3, 4, 5, 6$ is each $\frac{1}{6}$ or approximately $16.67\%$

Prize Wheel

Suppose a fair spinner is divided into $8$ equal sections, numbered $1$ through $8$. What is the probability that it stops on $5$?

• Possible Outcomes: $\{1, 2, 3, 4, 5, 6, 7, 8\}$

• Total Number of Outcomes: There are $8$ sections, so $k=8$

• Probability Mass Function:

  $$P(X=x) = \frac{1}{8}, \qquad x \in \{1,2,3,4,5,6,7,8\}$$

• Meaning: The probability that the wheel stops on number $5$ (or any other number) is $\frac{1}{8}$ or $12.5\%$

Exercises

1. A bag contains $10$ identical balls except for their colors: $1 \text{ red ball}$, $1 \text{ blue ball}$, $1 \text{ green ball}$, and so on up to $10$ different colors. If one ball is drawn randomly, what is the probability of drawing a yellow ball? Also write the probability mass function.

2. In a standard bridge card set containing $52$ cards, each card has the same probability of being drawn. What can you conclude about the probability distribution of drawing one card from that deck? What is the probability of drawing the King of Hearts?

Answer Key

1. Answer for Colored Ball Problem

   • Step $1$: Calculate total possibilities.

     There are $10$ balls with different colors, so the total possible outcomes is $k=10$.

   • Step $2$: Determine the probability mass function.

     Because the balls are identical except for color, each one has the same probability of being drawn. Use the discrete uniform formula:

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

     Here $x$ represents one of the $10$ available ball colors.

   • Step $3$: Calculate the probability of the yellow ball.

     The yellow ball is one of the $10$ balls, so its probability is the same as that of every other ball:

     $$P(\text{Yellow Ball}) = \frac{1}{10}$$

     Therefore, the probability of drawing a yellow ball is $\frac{1}{10}$ or $10\%$

2. Answer for Bridge Card Problem

   • Step $1$: Analyze the type of distribution.

     Since every card has the same probability of being drawn, the probability distribution is a discrete uniform distribution.

   • Step $2$: Calculate total possibilities.

     A standard bridge card set has $52$ different cards, so $k=52$

   • Step $3$: Calculate the probability of King of Hearts.

     The King of Hearts is one of the $52$ available cards, so using the uniform distribution principle:

     $$P(\text{King of Hearts}) = \frac{1}{52}$$

     Therefore, the probability of drawing the King of Hearts is $\frac{1}{52}$ or approximately $1.92\%$