B. Probability Distributions · Discrete Probability Distributions

Discrete Probability Distributions: Unit Review

B1.1 B1.2 B1.3 B1.4 B1.5 B1.6 B1.7About 40 minutes

Learning goals

I can build and interpret discrete probability distributions and their histograms.
I can find and use expected values, including in games and decisions.
I can choose and use the uniform, binomial and hypergeometric models, and simulate them.

Words to know

expected value
E(X) = Σ x·P(X = x), the long-run average.

binomial distribution
Successes in n independent trials with constant p: C(n, k)pᵏqⁿ⁻ᵏ.

hypergeometric distribution
Successes in a sample of n drawn without replacement from N items with r successes.

Let's review!

In this unit you turned outcomes into numbers with discrete random variables, organized them into probability distributions, and drew probability histograms. You found expected values and used them to judge games, raffles, lotteries and real decisions. You met three important models, the uniform, binomial and hypergeometric distributions, compared them, simulated them, and used them to solve problems such as hotel cancellations, overbooking and quality control.

The sibling unit Continuous Distributions and the Normal Distribution picks up from here: binomial and hypergeometric distributions with large n become bell-shaped, and the normal distribution models them. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work one or two sections at a time with a calculator (and a spreadsheet if you like). For each question, first name the distribution and its parameters (n, p, or N, r, n), then write the values of X you need ("at least 3" = 3, 4, ...). Keep 4 decimal places until the end. For every question you miss, write down why (wrong model? off-by-one? forgot the cost to play?) and go back to that lesson. Finish with the 10-question quiz.

The whole unit on one page: three models, their formulas and their means, and the rules that hold for every discrete distribution.

Just for fun

You have now earned the right to answer "What are the odds?" with "Which distribution are we using?" Your friends may find this less exciting than you do.

Discrete Random Variables and Probability Distributions

Classify as discrete or continuous: (a) the number of maple trees on a property; (b) the amount of sap one tree produces in a season.

P(X = 1) = 0.25, P(X = 2) = 0.4, P(X = 3) = k. Find k and P(X ≥ 2).

Two dice are rolled and X is the sum. Find P(X = 9) and P(X > 9).

Use this histogram for the questions in the next two sections.

Probability Histograms and Frequency Histograms

Use the review histogram (bars 0.10, 0.25, 0.30, 0.20, 0.15 at x = 1 to 5). Find the mode and P(X ≥ 4).

Why do the bar areas in a probability histogram add to 1?

In 80 trials, a value occurred 18 times. What is its relative frequency? Is it a probability or an estimate?

Expected Value: The Long-Run Average

Find E(X) for the review histogram (bars 0.10, 0.25, 0.30, 0.20, 0.15 at x = 1 to 5).

E(X) = 3.05. Find E(4X − 2).

A spinner gives 1 point (prob. 0.5), 4 points (0.3) and k points (0.2). E(X) = 3.3. Find k.

Expected Value in Games, Raffles and Lotteries

A game costs $3. Roll two dice: a sum of 12 pays $50 (total); anything else pays nothing. Find E(net).

What prize (total) would make the game in the previous question fair?

3000 tickets at $10; prizes $5000, $1000 and five $200. Find the expected net gain of one ticket.

The Discrete Uniform Distribution

X is a random integer from 12 to 47 inclusive. Find n and E(X).

For the same X, find P(20 ≤ X ≤ 29).

A spinner has 4 equal sectors: 3, 3, 6, 20. Is X uniform? Find E(X).

Binomial Experiments and the Binomial Probability Formula

List the four conditions for a binomial experiment.

X ~ B(7, 0.4). Find P(X = 3).

A die is rolled 5 times. Find P(at least one 6).

Is "number of aces in a 4-card hand from one deck" binomial? Explain.

The Binomial Distribution: Tables, Graphs and Expected Value

X ~ B(30, 0.2). Find E(X) and σ.

X ~ B(4, 0.5). Find P(X ≤ 1).

Describe the shape of B(20, 0.85).

The Hypergeometric Distribution: Dependent Trials

A box of 12 donuts has 5 maple-glazed. You take 3 at random. Find P(exactly 1 maple-glazed).

For the donuts, find E(X).

10 people (6 adults, 4 youth) are in a draw for 4 tickets. Find P(all 4 winners are adults).

Comparing Discrete Distributions: Binomial and Hypergeometric

Draw 2 cards. Find P(both hearts) with and without replacement.

A sample of 15 is taken from 6000 items. Is the binomial a good approximation to the hypergeometric?

B(25, 0.5) has σ = 2.5. Find σ for B(100, 0.5) and describe how the shape changes.

Simulating Probability Distributions

How would you use random digits to simulate a trial with p = 0.6?

A simulation of 50 trials gives X = 0 (5), 1 (15), 2 (20), 3 (10). Estimate E(X).

Write a spreadsheet formula for a success with probability 0.12.

Solving Problems with Uniform, Binomial and Hypergeometric Distributions

Name the model: "10 shots by a 75% shooter; X = makes."

A hotel has 12 reservations; each is cancelled with probability 0.1, independently. Find P(no cancellations) and P(at least 2).

A batch of 40 has 4 defective items. A sample of 6 is accepted if it has no defectives. Find P(accept).

Making Decisions with Expected Value

Option A: $600 with probability 0.4, else −$100. Option B: $150 for sure. Compare expected values.

A 4-choice test scores +1 right and −0.5 wrong. Should you guess at random?

Pools of 8 samples, each contaminated with probability 0.05. Find the expected number of tests per pool (1 test if negative, 9 if positive).

Check your understanding

Quiz

Which is a discrete random variable?

P(X = 0) = 0.2, P(X = 1) = 0.5, P(X = 2) = 0.3. What is E(X)?

A game pays $20 (total) with probability 0.1 and costs $3. What is the expected net gain?

X is a random integer from 1 to 60. What is P(X ≤ 15)?

X ~ B(12, 0.25). What is E(X)?

Which situation is hypergeometric?

X ~ B(5, 0.5). What is P(X = 2)?

Compared with sampling with replacement, sampling without replacement gives...

To simulate p = 0.07 with two-digit random numbers 00 to 99, success could be...

10.

Option A: $1000 with probability 0.25, else $0. Option B: $240 for sure. Which is true?

For parents and teachers

Use the review to find what to revisit:

Discrete vs continuous mix-ups, or tables that don't add to 1 → Lesson 1 (Discrete Random Variables and Probability Distributions).
Confusing frequency and probability histograms, or misreading skew → Lesson 2 (Probability Histograms and Frequency Histograms).
Averaging values without weighting by probability, or E(aX + b) errors → Lesson 3 (Expected Value: The Long-Run Average).
Forgetting the cost to play, or misjudging fair games and lotteries → Lesson 4 (Expected Value in Games, Raffles and Lotteries).
Off-by-one counting in a range, or using (a + b)/2 for uneven values → Lesson 5 (The Discrete Uniform Distribution).
Missing the C(n, k) factor, or not checking the binomial conditions → Lesson 6 (Binomial Experiments and the Binomial Probability Formula).
"At least"/"at most" errors, or np and √(npq) mix-ups → Lesson 7 (The Binomial Distribution: Tables, Graphs and Expected Value).
Using the binomial when sampling without replacement → Lesson 8 (The Hypergeometric Distribution: Dependent Trials).
Not sure when the models agree or how n changes the shape → Lesson 9 (Comparing Discrete Distributions: Binomial and Hypergeometric).
Assigning the wrong number of random digits, or trusting too few trials → Lesson 10 (Simulating Probability Distributions).
Choosing the wrong model in word problems → Lesson 11 (Solving Problems with Uniform, Binomial and Hypergeometric Distributions).
Leaving out outcomes or ignoring risk in decisions → Lesson 12 (Making Decisions with Expected Value).

Looking ahead: the next unit, Continuous Distributions and the Normal Distribution, moves from counting to measuring, introduces the normal distribution and z-scores, and uses the normal distribution to approximate binomial and hypergeometric probabilities.