Discrete and Binomial Probability Distributions: Concepts, Calculations, and Applications

Section 6.1 Discrete Random Variables

Learning Objectives

• Distinguish between discrete and continuous random variables

• Identify discrete probability distributions

• Graph discrete probability distributions

• Compute and interpret the mean of a discrete random variable

• Interpret the mean of a discrete random variable as an expected value

• Compute the standard deviation of a discrete random variable

Random Variables

A random variable is a numerical measure of the outcome from a probability experiment, determined by chance. Random variables are typically denoted by capital letters such as X.

Types of Random Variables

• Discrete random variable: Has a finite or countable number of values. These values can be plotted on a number line with spaces between each point (e.g., 0, 1, 2, 3, 4).

• Continuous random variable: Has infinitely many values, which can be plotted on a line in an uninterrupted fashion (e.g., all real numbers between 0 and 4).

Examples

• Discrete: The number of A's earned in a class of 15 students (X = 0, 1, ..., 15); the number of cars passing through a drive-through in an hour (X = 0, 1, 2, ...).

• Continuous: The speed of the next car passing a state trooper (S > 0, any positive real number).

Discrete Probability Distributions

The probability distribution of a discrete random variable X provides the possible values of X and their corresponding probabilities. This can be represented as a table, graph, or mathematical formula.

Example: Discrete Probability Distribution

Suppose a basketball player shoots three free throws. Let X be the number of shots made. The probability distribution is:

x	P(x)
0	0.01
1	0.10
2	0.38
3	0.51

We denote probabilities as P(x), read as "the probability that the random variable X equals x."

Rules for a Discrete Probability Distribution

• $\sum P(x) = 1$

• $0 \leq P(x) \leq 1$ for all x

Identifying Valid Distributions

To determine if a table is a valid discrete probability distribution, check that all probabilities are between 0 and 1, and that their sum is 1.

x	P(x)	Valid?
1	0.20
2	0.35
3	0.12
4	0.40
5	-0.07	No (negative probability)

Additional info: Probabilities must be non-negative and sum to 1.

Graphing Discrete Probability Distributions

Discrete probability distributions can be graphed as histograms, where the x-axis represents the values of the random variable and the y-axis represents the probabilities.

Mean (Expected Value) of a Discrete Random Variable

The mean (or expected value) of a discrete random variable X is given by:

$\mu_X = \sum [x \cdot P(x)]$

where x is a value of the random variable and P(x) is its probability.

Example Calculation

Using the basketball free throw example:

x	P(x)	x·P(x)
0	0.01	0.00
1	0.10	0.10
2	0.38	0.76
3	0.51	1.53

Sum: $0.00 + 0.10 + 0.76 + 1.53 = 2.39$

So, $\mu_X = 2.39$

Interpretation

If an experiment is repeated many times, the average value of X will approach the mean $\mu_X$.

$\overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$

As $n$ increases, $\overline{x}$ approaches $\mu_X$.

Expected Value

The mean of a random variable is also called its expected value, denoted $E(X)$.

Example: Expected Value in Insurance

Suppose an 18-year-old male buys a $250,000 1-year term life insurance policy for $350. The probability of surviving the year is 0.998937, and dying is 0.001063.

x	P(x)
$350$ (survives)	0.998937
-$249,650$ (dies)	0.001063

Expected value to the company:

$E(X) = 350(0.998937) + (-249,650)(0.001063) = 84.25$

Interpretation: On average, the company expects to make $84.25 per policy in the long run.

Standard Deviation of a Discrete Random Variable

The standard deviation of a discrete random variable X measures the spread of its probability distribution and is given by:

$\sigma_X = \sqrt{\sum (x - \mu_X)^2 P(x)}$

Alternatively,

$\sigma_X = \sqrt{\sum x^2 P(x) - \mu_X^2}$

Example Calculation

x	P(x)	(x - μ)2P(x)
0	0.01	0.057121
1	0.10	0.193209
2	0.38	0.057879
3	0.51	0.189791

Sum: $0.057121 + 0.193209 + 0.057879 + 0.189791 = 0.4979$

$\sigma_X = \sqrt{0.4979} \approx 0.7$

Using Technology

Statistical software or graphing calculators can be used to compute the mean and standard deviation for a discrete probability distribution.

Section 6.2 The Binomial Probability Distribution

Learning Objectives

• Determine whether a probability experiment is a binomial experiment

• Compute probabilities of binomial experiments

• Compute the mean and standard deviation of a binomial random variable

• Graph a binomial probability distribution

Binomial Probability Distribution

The binomial probability distribution is a discrete probability distribution for experiments with two mutually exclusive outcomes (success or failure) in each trial.

Criteria for a Binomial Experiment

• The experiment is performed a fixed number of times (n trials).

• The trials are independent.

• Each trial has two mutually exclusive outcomes: success or failure.

• The probability of success (p) is the same for each trial.

Notation

• n: Number of independent trials

• p: Probability of success on a single trial

• 1-p: Probability of failure

• X: Number of successes in n trials (possible values: 0, 1, ..., n)

Examples

• Binomial: Basketball player shoots 3 free throws (n=3, p=0.8, X=number made)

• Not binomial: Drawing 3 cards from a deck without replacement (trials are not independent)

Binomial Probability Distribution Function

The probability of obtaining x successes in n independent trials is:

$P(x) = {n \choose x} p^x (1-p)^{n-x}$

where ${n \choose x}$ is the binomial coefficient.

Key Phrases and Mathematical Symbols

Phrase	Symbol
Exactly x	$X = x$
At least x	$X \geq x$
No more than x	$X \leq x$
Fewer than x	$X < x$

Example: Calculating Binomial Probabilities

• Given n = 10, p = 0.72, find the probability that exactly 8 would rather give up chocolate:

$P(8) = {10 \choose 8} (0.72)^8 (0.28)^2 = 0.2548$

• Probability that fewer than 3 would rather give up chocolate:

$P(X < 3) = P(0) + P(1) + P(2) = 0.000003 + 0.000076 + 0.000881 = 0.00096$

• Probability that at least 3 would rather give up chocolate:

$P(X \geq 3) = 1 - P(X < 3) = 0.9990$

• Probability that the number is between 5 and 7, inclusive:

$P(5 \leq X \leq 7) = P(5) + P(6) + P(7) = 0.0839 + 0.1798 + 0.2642 = 0.5279$

Using Binomial Tables and Technology

Binomial tables and statistical software (e.g., StatCrunch, TI-84 calculator) can be used to find binomial probabilities efficiently, especially for larger n.

Mean and Standard Deviation of a Binomial Random Variable

For a binomial experiment with n trials and probability of success p:

• Mean (expected value): $\mu_X = np$

• Standard deviation: $\sigma_X = \sqrt{np(1-p)}$

Example

In a sample of 300, with p = 0.72:

$\mu_X = 300 \times 0.72 = 216$

$\sigma_X = \sqrt{300 \times 0.72 \times 0.28} = 7.8$

Interpretation: Expect about 216 successes, with a standard deviation of 7.8.

Graphing Binomial Probability Distributions

Binomial distributions can be graphed for various values of n and p:

• For small p, the distribution is skewed right.

• For p = 0.5, the distribution is symmetric and bell-shaped.

• For large p, the distribution is skewed left.

As n increases, the distribution becomes more bell-shaped, especially if $np(1-p) \geq 10$.

Using the Empirical Rule

To check for unusual results, calculate $\mu_X \pm 2\sigma_X$. Values outside this range are considered unusual.

Example: For $\mu_X = 216$, $\sigma_X = 7.8$, the range is $200.4$ to $231.6$. Observing 230 is not unusual.