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Prealgebra / LEVEL 9 · DIFFICULTY 3/5

Counting & Probability

Describe a sample space and count possibilities without omission or duplication.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST 9.1Addition, Multiplication & ComplementsRead the lesson
  2. MINI QUEST 9.2Casework, Pairs & Systematic ListsRead the lesson
  3. MINI QUEST 9.3Probability & Equally Likely OutcomesRead the lesson
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STAGE 9.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Addition, Multiplication & Complements

Goal: Choose whether cases should be added or stages multiplied.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Add disjoint alternatives

If exactly one of several nonoverlapping cases occurs, add their counts. If cases overlap, subtract the overlap so each object is counted once.

|A ∪ B| = |A| + |B| − |A ∩ B|

Multiply successive choices

If every first choice has the same number of available second choices, multiply the counts. With restrictions, update the choices at each stage.

m first choices × n choices each = mn outcomes

Count a complement when it is simpler

Count all possibilities and remove those that violate the requirement. Define the universe clearly before subtracting.

wanted = total − unwanted
WORKED EXAMPLE 1

A menu has 3 soups and 4 sandwiches. Choose one of each. How many meals?

  1. Each soup pairs with four sandwiches.
  2. Three groups of four give the count.

12

WORKED EXAMPLE 2

Among 30 students, 18 play piano, 16 play guitar and 9 play both. How many play neither?

  1. The union is 18 + 16 − 9 = 25.
  2. Subtract from all 30 students.

5

Common pitfalls

Possible mix-up: Always multiply counts appearing in a problem.

Multiply stages; add disjoint alternatives.

Possible mix-up: Subtracting overlap is unnecessary.

Objects in both sets were included in each count and need one duplicate removed.

Explain it to yourself

Give an example where adding two group counts would double-count some people.

Preview the eight practice prompts
  1. Choose one of 4 shirts and one of 3 hats. How many outfits are possible?
  2. A snack is either one of 5 fruits or one of 4 yogurts. Choose exactly one snack. How many choices?
  3. A three-symbol code uses A, B or C in each position, with repetition allowed. How many codes exist?
  4. A two-symbol code uses distinct letters from A, B, C, D, E. How many ordered codes exist?
  5. Of 40 students, 22 like art, 19 like music and 8 like both. How many like at least one?
  6. How many two-digit positive integers are not multiples of 10?
  7. A cafe offers 3 breads, 4 fillings and 2 sauces. One filling is unavailable, and every remaining combination is allowed. How many sandwiches can be made? New context
  8. A password has four positions, each either 0 or 1. How many passwords contain at least one 1? New context
Open stage 9.1 in the student workspace →

STAGE 9.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Casework, Pairs & Systematic Lists

Useful preparation: Addition, Multiplication & Complements

Goal: Choose a classification that counts each possibility exactly once.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Choose a nonoverlapping case label

A useful case label can be the first digit, a smallest element or the number of a certain type. Check that every outcome fits exactly one case.

complete + disjoint cases ⇒ valid sum

Order changes what counts as different

Ordered pairs distinguish (A,B) from (B,A). Unordered selections treat them as one pair. When dividing an ordered count, every unordered outcome must have the same number of orderings.

unordered pairs from n objects = n(n − 1)/2

Use bounds to make a finite list

Translate a condition into a range before listing. Tables and trees provide a record that exposes missing or repeated possibilities.

classify, bound, list, check
WORKED EXAMPLE 1

Six friends each shake hands once with every other friend. How many handshakes?

  1. There are 6 × 5 ordered person-partner choices.
  2. Every handshake appears twice.

15

WORKED EXAMPLE 2

How many nonnegative integer pairs (x,y) satisfy x + y = 4?

  1. List x = 0, 1, 2, 3, 4.
  2. Each x determines exactly one y.

5

Common pitfalls

Possible mix-up: Divide every ordered count by 2.

The appropriate divisor depends on how many representations each outcome has.

Possible mix-up: A long list guarantees completeness.

The list needs an organizing rule that covers every possibility.

Explain it to yourself

Why does choosing a smallest element often prevent double-counting?

Preview the eight practice prompts
  1. How many unordered pairs can be chosen from 7 distinct objects?
  2. How many nonnegative integer pairs (x,y) satisfy x + y = 8?
  3. How many positive integer pairs (x,y) satisfy x + y = 8?
  4. Using digits 1, 2, 3, 4 with no repetition, how many two-digit even numbers can be formed?
  5. How many unordered pairs of distinct positive integers have sum 11?
  6. How many rectangles can be formed by choosing edges from a grid with 3 vertical lines and 4 horizontal lines?
  7. Eight clubs each play every other club exactly once. How many matches are scheduled? New context
  8. A booth sells $2 and $5 tokens. How many nonnegative integer pairs (a,b) satisfy 2a + 5b = 20? New context
Open stage 9.2 in the student workspace →

STAGE 9.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Probability & Equally Likely Outcomes

Useful preparation: Casework, Pairs & Systematic Lists · Combining & Scaling Fractions

Goal: Count favorable outcomes in a justified sample space.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Specify the random process

A probability model needs the possible outcomes and their weights. Counting favorable divided by total works directly only when outcomes are equally likely.

P(event) = favorable / total for equal weights

Complementary events exhaust the sample space

An event and its complement are disjoint and together cover all outcomes. Their probabilities sum to one.

P(not A) = 1 − P(A)

Update after a draw without replacement

Removing an object changes both total count and category counts. For a sequence, multiply each conditional probability using the updated situation.

P(A then B) = P(A) × P(B after A)
WORKED EXAMPLE 1

A fair die is rolled. Find the probability of a number greater than 4.

  1. There are six equally likely outcomes.
  2. Only 5 and 6 qualify.

1/3

WORKED EXAMPLE 2

A bag has 3 red and 2 blue tokens. Two are drawn without replacement. Find the probability both are blue.

  1. First blue has probability 2/5.
  2. Then one blue remains among four tokens.

(2/5)(1/4) = 1/10

Common pitfalls

Possible mix-up: Any listed outcomes are equally likely.

Equal likelihood must follow from the random process.

Possible mix-up: Without replacement, use the same denominator for every draw.

The total remaining objects decreases after a draw.

Explain it to yourself

How would replacing the first token change a two-draw calculation?

Preview the eight practice prompts
  1. Roll a fair six-sided die numbered 1–6. What is the probability of an even number?
  2. A bag has 4 red and 6 blue equally selectable tokens. What is the probability of drawing red?
  3. A fair coin is tossed three times. What is the probability of all heads?
  4. A fair die is rolled. What is the probability of not rolling 6?
  5. Two fair dice are rolled independently. What is the probability their sum is 7?
  6. A bag has 5 green and 3 yellow tokens. Draw two without replacement. What is the probability both are green?
  7. A game uses a fair coin twice. You win if at least one toss is heads. What is the winning probability? New context
  8. A box has 2 red, 3 blue and 5 white tokens, each equally likely. One is drawn. What is the probability it is red or blue? New context
Open stage 9.3 in the student workspace →
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