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Discrete Probability Distributions Reference cheat sheet - grade 11-12

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Math Grade 11-12

Discrete Probability Distributions Reference Cheat Sheet

A printable reference covering PMFs, CDFs, expected value, variance, Bernoulli, binomial, geometric, Poisson, and hypergeometric distributions for grades 11-12.

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Discrete probability distributions describe random variables that take countable values, such as the number of successes, arrivals, or defective items. This cheat sheet helps students identify common discrete models and use their formulas correctly. It is useful for probability units, statistics courses, exam review, and quick reference during problem solving.

The most important ideas are the probability mass function, the cumulative distribution function, expected value, and variance. Common distributions include Bernoulli, binomial, geometric, Poisson, and hypergeometric models. Choosing the correct distribution depends on whether trials are independent, whether probabilities stay constant, and whether sampling is done with or without replacement.

Key Facts

  • A probability mass function satisfies P(X=x)≥0P(X=x) \ge 0 for every value xx and ∑xP(X=x)=1\sum_x P(X=x)=1.
  • The cumulative distribution function is F(x)=P(X≤x)F(x)=P(X \le x), so for discrete variables P(a≤X≤b)=F(b)−F(a−1)P(a \le X \le b)=F(b)-F(a-1) when values are integers.
  • The expected value of a discrete random variable is E(X)=∑xxP(X=x)E(X)=\sum_x xP(X=x).
  • The variance is Var⁡(X)=E(X2)−[E(X)]2\operatorname{Var}(X)=E\left(X^2\right)-\left[E(X)\right]^2, where E(X2)=∑xx2P(X=x)E\left(X^2\right)=\sum_x x^2P(X=x).
  • For a Bernoulli random variable with success probability pp, P(X=1)=pP(X=1)=p, P(X=0)=1−pP(X=0)=1-p, E(X)=pE(X)=p, and Var⁡(X)=p(1−p)\operatorname{Var}(X)=p(1-p).
  • For a binomial random variable X∼Bin⁡(n,p)X \sim \operatorname{Bin}(n,p), P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom{n}{k}p^k(1-p)^{n-k}, E(X)=npE(X)=np, and Var⁡(X)=np(1−p)\operatorname{Var}(X)=np(1-p).
  • For a geometric random variable counting trials until the first success, P(X=k)=(1−p)k−1pP(X=k)=(1-p)^{k-1}p, E(X)=1pE(X)=\frac{1}{p}, and Var⁡(X)=1−pp2\operatorname{Var}(X)=\frac{1-p}{p^2}.
  • For a Poisson random variable X∼Pois⁡(λ)X \sim \operatorname{Pois}(\lambda), P(X=k)=e−λλkk!P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}, E(X)=λE(X)=\lambda, and Var⁡(X)=λ\operatorname{Var}(X)=\lambda.

Vocabulary

Discrete random variable
A random variable whose possible values can be listed, such as 0,1,2,30,1,2,3 or another countable set.
Probability mass function
A function that gives P(X=x)P(X=x) for each possible value xx of a discrete random variable.
Cumulative distribution function
A function defined by F(x)=P(X≤x)F(x)=P(X \le x) that gives the probability of being at or below a value.
Expected value
The long-run average value of a random variable, calculated by E(X)=∑xxP(X=x)E(X)=\sum_x xP(X=x).
Variance
A measure of spread around the mean, calculated by Var⁡(X)=E(X2)−[E(X)]2\operatorname{Var}(X)=E\left(X^2\right)-\left[E(X)\right]^2.
Independent trials
Trials are independent when the outcome of one trial does not change the probabilities for the others.

Common Mistakes to Avoid

  • Using the binomial model when probabilities change, which is wrong because X∼Bin⁡(n,p)X \sim \operatorname{Bin}(n,p) requires independent trials with the same success probability pp.
  • Forgetting the combination factor in binomial probability, which is wrong because (nk)\binom{n}{k} counts the different orders in which kk successes can occur.
  • Confusing P(X=k)P(X=k) with P(X≤k)P(X \le k), which is wrong because a PMF gives one exact value while a CDF adds probabilities up to that value.
  • Using the geometric formula with the wrong starting point, which is wrong because P(X=k)=(1−p)k−1pP(X=k)=(1-p)^{k-1}p applies when XX counts the trial number of the first success.
  • Assuming variance equals standard deviation, which is wrong because standard deviation is σ=Var⁡(X)\sigma=\sqrt{\operatorname{Var}(X)}.

Practice Questions

  1. 1 A fair coin is flipped 88 times. If XX is the number of heads, find P(X=5)P(X=5).
  2. 2 A call center receives an average of 33 calls per minute. Using a Poisson model, find P(X=0)P(X=0) for one minute.
  3. 3 A quiz has 44 multiple-choice questions with 55 choices each. If a student guesses randomly, find E(X)E(X) for the number of correct answers.
  4. 4 A box contains 66 red marbles and 44 blue marbles, and 33 marbles are drawn without replacement. Explain why a hypergeometric model is more appropriate than a binomial model.

Understanding Discrete Probability Distributions Reference

A distribution starts with the possible values of a count, called its support. Writing this list first prevents many mistakes. A variable counting faulty bulbs in a box of five cannot be negative or larger than five.

A probability table shows how likely each allowed value is. The probabilities describe single outcomes, while the cumulative function collects probability as values are included from left to right. For a discrete variable, this graph has flat sections and jumps.

The size of each jump is the probability at that value. This makes cumulative graphs useful for finding chances of ranges and checking whether a table has been entered correctly.

Expected value is a long-run average, not a promise about one result. A game with an expected value of three points can still give zero points on many turns. Each outcome affects the average according to both its size and its probability.

Rare large outcomes can pull the expected value upward. Variance measures spread around that average. A larger variance means results are less predictable.

Its units are squared, so standard deviation is often easier to interpret because it uses the same units as the original count. When comparing two plans with the same expected result, the one with lower spread is usually more consistent.

The hardest part of these problems is often choosing a model. Use a Bernoulli model for one yes or no outcome, such as whether a selected item is defective. Use a binomial model only when there is a fixed number of trials, each trial has two outcomes, the success chance stays the same, and trials are independent.

A geometric model continues until the first success. Read its definition carefully. Some books count the trial of the first success, while others count failures before it.

A hypergeometric model fits a sample from a finite group without replacement. Each selection changes the group left behind, so independence does not hold. A Poisson model is useful for counts during a fixed time or space interval, such as calls arriving in a minute or flaws along a length of wire.

Real data rarely follows a model perfectly. Arrivals may cluster at busy times. Products from one machine may share the same defect.

Survey answers can influence one another. These details can make a simple distribution inaccurate. Before calculating, state what one trial means, what counts as success, and what values the variable can take.

Then check whether the answer is sensible. A probability must fall from zero to one. A count probability outside its possible range must be zero.

For binomial work, distinguish exactly from at least and at most. These phrases change which probabilities must be added. Technology can calculate long sums, but it cannot decide whether the assumptions match the situation.