Why this chapter matters
Discrete models describe counts and repeated yes-no processes that appear in reliability, quality, and survey settings.
What you will learn
- Construct and interpret probability mass functions.
- Calculate expected value and standard deviation for a discrete variable.
- Use binomial and geometric models under their assumptions.
Understand the core ideas
Discrete random variables model countable outcomes, such as number of successes in fixed trials or number of attempts before first success. The probability mass function assigns probabilities to each possible value, and those probabilities must be nonnegative and sum to one. Expected value gives a long run average over repeated trials, while variance and standard deviation measure spread around that average. These summaries describe distribution behavior, not a guaranteed single outcome.
Model choice matters. A binomial model applies when trial count is fixed, each trial has two outcomes, trials are independent, and success probability stays constant. A geometric model applies when you count trials until first success under the same constant success chance. Using the wrong model changes formulas and interpretations. Distinguishing discrete and continuous variables also prevents a common mistake: discrete models can assign probability to exact values, while continuous models assign probability to intervals. Clear model labels also make review and debugging easier when teams share analyses.
Key terms
- Probability mass function
- A table or rule giving probability for each discrete variable value.
- Expected value
- The long run average value across many repetitions of the process.
- Binomial random variable
- Count of successes in a fixed number of independent Bernoulli trials.
- Geometric random variable
- Number of trials needed to get the first success.
Binomial count of heads
Let X be number of heads in 4 fair coin flips, so X follows binomial with n = 4 and p = 0.5.
- Identify target probability as P(X = 2).
- Use binomial formula C(4,2) * * .
- Compute C(4,2) = 6 and probability 6/16.
- Convert to decimal 0.375 and interpret in context.
A common misconception
Claim: Expected value must be one of the possible observed outcomes.
Correction: Expected value is a long run average and can be a non-attainable number for individual trials.
Lessons in this chapter
- Random variables and distribution tablesTranslate outcome rules into a valid probability distribution. Read the full guide →
- Expected value and variabilityCompute mean and variance from a discrete distribution.
- Binomial modelIdentify fixed-trial Bernoulli settings and compute binomial probabilities.
- Geometric modelModel the number of trials until first success and interpret long-run behavior.
Study task
Chapter checkpoint
What conditions justify a binomial model?
A fixed number of trials, independent trials, only success or failure outcomes, and a constant success probability across trials.