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Virginia SOL Mathematics Textbook

Grade 8 Workbook — Chapter 15: Probability of Independent and Dependent Events

SOL 8.PS.1 · Companion to Textbook Chapter 15

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 130.

PAGE 1 — Chapter opener

Chapter 15 · Probability of Independent and Dependent Events

Standard 8.PS.1

In this chapter you will:

Words to know: outcome · sample space · event · compound event · independent events · dependent events · with replacement · without replacement · tree diagram · multiplication rule

The one question this chapter asks:

After the first draw, is the collection the same as it was?

Not in this chapter: three-stage problems. Every problem on these pages has exactly two events.

PAGE 2 — Replacement or not

15.1 Independent or Dependent: What Replacement Changes

FIGURE: fig1-replacement-or-not.png (full width)

Fill in the blanks.

Two events are independent when the first does ____________ change the probability of the second.

Two events are dependent when the first ____________ change the probability of the second.

With replacement: the first item is ____________, so the collection is ____________.

Without replacement: the first item is ____________, so the total drops by ____________ and the count of that kind drops by ____________.

Use the bag of 3 red and 2 blue marbles above.

  1. Marbles in the bag before the first draw: __________

  2. Marbles the second draw comes from, with replacement: __________

  3. Marbles the second draw comes from, without replacement: __________

  4. First marble is red and kept out. Red left: ______ Blue left: ______ Total: ______

  5. First marble is blue and kept out. Red left: ______ Blue left: ______ Total: ______

PAGE 3 — Naming the case

Independent or Dependent?

  1. A coin is flipped, then a die is rolled. ______________________

  2. A marble is drawn and kept, then a second is drawn. ______________________

  3. In one sentence, state what replacement does to the collection the second event draws from.

Wording table — complete the right-hand column.

What the problem says Independent or dependent?
"with replacement" / "puts it back"
"without replacement" / "keeps it"
"draws two at the same time"
two different experiments (coin and die)

  1. Classify each pair of events. Write I for independent or D for dependent.

    a) A spinner is spun twice. ______

    b) Two cards are dealt, the first not returned. ______

    c) A die is rolled, then a coin is flipped. ______

    d) A name is drawn from a hat, returned, and a name is drawn again. ______

    e) Two socks are pulled from a drawer and both kept out. ______

PAGE 4 — Rebuilding the second collection

What Is Left After the First Draw?

A bag holds 4 white and 6 black tiles.

  1. A white tile is drawn and replaced.

    White: ______ Black: ______ Total: ______

  2. A white tile is drawn and kept out.

    White: ______ Black: ______ Total: ______

  3. A black tile is drawn and kept out.

    White: ______ Black: ______ Total: ______

  4. A bag holds 7 lettered tiles; two are drawn without replacement.

    First denominator: ______ Second denominator: ______

  5. Two cards are drawn from a deck of 52.

    With replacement: ______ and ______ Without replacement: ______ and ______

  6. Explain why putting the first item back makes the two events independent.

  7. A jar holds 5 red and 5 green marbles. One is drawn and kept out.

    If the first was red, P(second is red)=P(\text{second is red}) = ____________

    If the first was green, P(second is red)=P(\text{second is red}) = ____________

    Which is larger? ____________ Why does this make the events dependent?

PAGE 5 — Applications and reasoning

Lesson 15.1 Practice

  1. Application. A box holds 12 buttons: 5 striped and 7 solid. A student draws one, drops it on the floor, and does not put it back before drawing again. Independent or dependent? Explain.

  2. Application. At a carnival you spin a red spinner and then a separate blue spinner. Independent or dependent? Explain what makes them so.

  3. Reasoning. Can two events be dependent when nothing is drawn from a bag? Describe a situation and explain.

  4. Error analysis. A student says, "The two draws are independent because both marbles are the same color." Explain the confusion and give the correct test for independence.

Exit ticket 15.1

  1. Define independent events in your own words, without using the word "independent."

  2. A bag holds 6 yellow and 4 purple marbles. Second-draw total with replacement: ______ Without: ______

  3. Same bag; first marble is yellow and kept out. Contents for the second draw: ____________________

  4. A game says, "Draw a chip, record its color, put it back, draw again." Independent or dependent, and how do you know?

PAGE 6 — Counting all the outcomes

15.2 The Probability of Two Independent Events

FIGURE: fig4-sample-space-table.png (full width)

Fill in.

A coin has ______ outcomes and a die has ______ outcomes, so together there are ______ ⋅\cdot ______ == ______ equally likely pairs.

Shaded outcomes (heads and even): ____________________

P(heads and even)=‾‾=‾P(\text{heads and even}) = \frac{\underline{\hspace{1cm}}}{\underline{\hspace{1cm}}} = \underline{\hspace{1.5cm}}

The rule for independent events.

P(A and B)=‾P(A \text{ and } B) = \underline{\hspace{3cm}}

Multiply, do not ____________.

FIGURE: fig5-area-model-product.png (half width, right)

The whole square is 11. The four areas add to ______.

PAGE 7 — The with-replacement tree

Reading a Tree Diagram

FIGURE: fig2-tree-with-replacement.png (full width)

Three checks for any tree.

  1. One coin flip and one roll of a die: P(heads and 5)=P(\text{heads and } 5) = ____________

  2. One coin flip and one roll of a die: P(tails and even)=P(\text{tails and even}) = ____________

  3. From the table on the previous page, count heads-and-even outcomes: ______ of ______ == ______. Does it match the rule? ______

  4. P(red and then red)=P(\text{red and then red}) = ______ ⋅\cdot ______ == ____________

  5. P(blue and then blue)=P(\text{blue and then blue}) = ______ ⋅\cdot ______ == ____________

  6. P(red and then blue)=P(\text{red and then blue}) = ______ ⋅\cdot ______ == ____________

  7. Sum of all four outcome probabilities: ____________ Why must it be this?

PAGE 8 — Two different experiments

Coin and Spinner

FIGURE: fig6-independent-coin-and-spinner.png (full width)

  1. P(heads and sector A)=P(\text{heads and sector } A) = ______ ⋅\cdot ______ == ____________

Why is this independent? The spinner branches under HH and under TT are ____________________.

Independent practice

  1. Two fair coins: P(heads and heads)=P(\text{heads and heads}) = ____________

  2. Two dice: P(both 6)=P(\text{both } 6) = ____________

  3. Two dice: P(both odd)=P(\text{both odd}) = ____________

  4. Spinner with 5 equal sectors, spun twice: P(3 and then 3)=P(3 \text{ and then } 3) = ____________

  5. Same spinner and a coin: P(even sector and tails)=P(\text{even sector and tails}) = ____________

  6. Bag of 3 red and 2 blue, with replacement: P(blue and then blue)=P(\text{blue and then blue}) = ____________

  7. Bag of 4 green and 6 yellow, with replacement: P(green and then yellow)=P(\text{green and then yellow}) = ____________

  8. Same bag and replacement: P(yellow and then yellow)=P(\text{yellow and then yellow}) = ____________ == ______ as a decimal

  9. A die and a 4-sector spinner: P(prime and sector C)=P(\text{prime and sector } C) = ____________

  10. A coin is flipped twice. P(no heads)=P(\text{no heads}) = ____________ so P(at least one head)=P(\text{at least one head}) = ____________

PAGE 9 — Applications

Lesson 15.2 Applications

  1. Application. A prize is in 16\frac{1}{6} of all cereal boxes, mixed at random. A shopper buys two boxes. P(both have a prize)=P(\text{both have a prize}) = ____________

    Work space:

  2. Application. Two unsynchronized traffic lights on separate roads are each green 25\frac{2}{5} of the time. P(both green)=P(\text{both green}) = ____________

    Work space:

  3. Reasoning. AA and BB are independent with P(A)=13P(A) = \frac{1}{3} and P(B)=12P(B) = \frac{1}{2}. Find P(A and B)P(A \text{ and } B) and explain why it must be smaller than both.

  4. Error analysis. A student writes 12+16=23\frac{1}{2} + \frac{1}{6} = \frac{2}{3} for P(heads and 5)P(\text{heads and } 5). Explain the error and give the correct probability.

Exit ticket 15.2

  1. P(heads and 3)P(\text{heads and } 3) for one flip and one roll: ____________

  2. A 4-sector spinner spun twice: P(sector A both times)=P(\text{sector } A \text{ both times}) = ____________

  3. Bag of 3 red and 2 blue with replacement: P(red and then red)=P(\text{red and then red}) = ______ == ______ (decimal) == ______ (percent)

  4. The rule for two independent events: P(A and B)=‾P(A \text{ and } B) = \underline{\hspace{3cm}}

PAGE 10 — The without-replacement tree

15.3 The Probability of Two Dependent Events

FIGURE: fig3-tree-without-replacement.png (full width)

The rule.

P(A and B)=P(A)⋅P(‾)P(A \text{ and } B) = P(A) \cdot P(\underline{\hspace{3cm}})

Two questions, in this order, before writing the second factor.

Compare the two trees.

With replacement Without replacement
First-stage branches
Second-stage denominators
Are the two sets of second-stage branches the same?

  1. P(red and then red)=P(\text{red and then red}) = ______ ⋅\cdot ______ == ____________

  2. P(red and then blue)=P(\text{red and then blue}) = ______ ⋅\cdot ______ == ____________

  3. P(blue and then blue)=P(\text{blue and then blue}) = ______ ⋅\cdot ______ == ____________

  4. Sum of all four: ____________ What does that tell you?

PAGE 11 — A larger bag

Ten Counters, Drawn Twice

FIGURE: fig7-dependent-tree-ten-counters.png (full width)

  1. P(green and then green)=P(\text{green and then green}) = ______ ⋅\cdot ______ == ____________

  2. P(yellow and then yellow)=P(\text{yellow and then yellow}) = ______ ⋅\cdot ______ == ____________

  3. P(green and then yellow)=P(\text{green and then yellow}) = ______ ⋅\cdot ______ == ____________

  4. Show that all four add to 11:

Watch out. Reducing the denominator and forgetting the numerator is the classic error. Say "two red out of four marbles" out loud before you write the fraction.

PAGE 12 — Independent practice

Lesson 15.3 Practice

A bag holds 5 red and 3 white marbles. Two are drawn WITHOUT replacement.

  1. P(both red)=P(\text{both red}) = ______ ⋅\cdot ______ == ____________

  2. P(both white)=P(\text{both white}) = ______ ⋅\cdot ______ == ____________

  3. P(red and then white)=P(\text{red and then white}) = ______ ⋅\cdot ______ == ____________

  4. P(white and then red)=P(\text{white and then red}) = ______ ⋅\cdot ______ == ____________

  5. Add items 59–62 and show the total is exactly 11:

  6. Two cards from a deck of 52, no replacement: P(both aces)=P(\text{both aces}) = ____________

  7. Two cards from a deck of 52, no replacement: P(both hearts)=P(\text{both hearts}) = ____________

  8. Drawer of 6 black and 4 blue socks: P(both black)=P(\text{both black}) = ____________

  9. Same drawer: P(both blue)=P(\text{both blue}) = ____________

  10. Same drawer: P(matching pair)=P(\text{matching pair}) = ______ ++ ______ == ____________

  11. A class of 12 girls and 8 boys; two different students chosen for two jobs. P(both boys)=P(\text{both boys}) = ____________

PAGE 13 — Applications and exit ticket

Lesson 15.3 Applications

  1. Application. A hat holds 15 raffle tickets, 3 of which win. You draw two and keep both. P(both win)=P(\text{both win}) = ____________

    Work space:

  2. Application. A bowl holds 4 apples and 5 oranges. Two are taken and eaten. P(apple then orange)=P(\text{apple then orange}) = ____________

    Work space:

  3. Error analysis. For 5 red and 3 white, a student writes P(both red)=58⋅48=516P(\text{both red}) = \frac{5}{8} \cdot \frac{4}{8} = \frac{5}{16}. Identify the error and give the correct probability.

Exit ticket 15.3

A jar holds 7 blue and 3 red chips, drawn twice without replacement.

  1. P(both blue)=P(\text{both blue}) = ____________

  2. P(both red)=P(\text{both red}) = ____________

  3. P(blue and then red)=P(\text{blue and then red}) = ____________

  4. In one sentence, what changes in the second factor when there is no replacement?

PAGE 14 — The comparison

15.4 Comparing and Contrasting the Two Cases

FIGURE: fig8-independent-versus-dependent.png (full width)

The same bag, both ways — complete the table. Bag: 3 red, 2 blue.

Outcome With replacement As a decimal Without replacement As a decimal
both red
red then blue
blue then red
both blue
total

The pattern. Without replacement, matching pairs get ____________ likely and mixed pairs get ____________ likely.

  1. P(both red)P(\text{both red}): with ______ without ______ Larger: ____________

  2. P(both blue)P(\text{both blue}): with ______ without ______ Larger: ____________

  3. P(red then blue)P(\text{red then blue}): with ______ without ______ Larger: ____________

PAGE 15 — Why the pattern happens

Explaining the Difference

  1. Why is drawing the same color twice less likely without replacement?

  2. Why is drawing two different colors more likely without replacement?

  3. Which factor of the product is the same in both computations, and why?

  4. Copy and complete: with replacement, the second denominator is ‾\underline{\hspace{2cm}}; without replacement, it is ‾\underline{\hspace{2cm}}.

A bag holds 4 green and 6 yellow counters. Compute each both ways and circle the larger.

  1. P(both green)P(\text{both green}): with ____________ without ____________

  2. P(both yellow)P(\text{both yellow}): with ____________ without ____________

  3. P(green then yellow)P(\text{green then yellow}): with ____________ without ____________

  4. Bag of 5 red and 3 white: P(both red)P(\text{both red}) with ____________ (≈\approx ______) without ____________ (≈\approx ______)

PAGE 16 — Applications

Lesson 15.4 Applications

  1. Name the case (independent or dependent), then compute.

    a) A coin flipped twice; both heads. Case: ______ P=P = ____________

    b) Two of 15 raffle tickets drawn and kept, 3 of the 15 win; both win. Case: ______ P=P = ____________

  2. Application. A game lets you draw two marbles from 3 red and 2 blue, with or without replacement, and you win with two reds. Better choice: ____________ By how much (decimal): ______

    Work space:

  3. Application. Same game and bag, but you win with one of each color in either order. Better choice: ____________ Why does the answer flip?

  4. Reasoning. A bag holds 1000 marbles, 500 red. P(both red)P(\text{both red}) with replacement: ______ Without (estimate): ______ Why so close?

  5. Application. Two cards from a deck of 52, both spades: with ____________ without ____________ Greater: ______

  6. Application. 25 name cards, 5 starred; two chosen. P(both starred)P(\text{both starred}) with ____________ without ____________

  7. Error analysis. "Independent and dependent probabilities use different operations: one multiplies and one adds." Correct the statement.

  8. Reasoning. Describe a two-draw situation where both answers are exactly equal, and explain why.

Exit ticket 15.4

  1. Drawer of 6 black and 4 blue socks: P(both black)P(\text{both black}) with ______ without ______ Larger: ______

  2. One similarity and one difference between the two rules:

  3. Two marbles are drawn "at the same time." Independent or dependent, and why? _________________

  4. How do you decide what the second denominator should be? _________________________________

PAGE 17 — Review Part A

Chapter 15 Review · Part A

Independent or dependent, and the role of replacement (8.PS.1a)

  1. Classify each. Write I or D.

a) Two spins of the same spinner. ______

b) Two marbles drawn, the first kept out. ______

c) A die rolled, then a coin flipped. ______

d) Two names drawn from a hat, the first returned. ______

e) Two cards dealt, both kept. ______

  1. Bag of 5 red, 4 green, 1 white. A green marble is drawn and replaced. Contents: ______________ Total: ______

  2. Same bag. A green marble is drawn and kept out. Contents: ______________ Total: ______

  3. In two sentences, explain how replacement impacts the probability of the second event. Mention the numerator and the denominator.

_________________________________________________________________

_________________________________________________________________

  1. Two cards dealt from 52 without replacement. Second denominator: ______

  2. Two dice are rolled. Independent? ______ Explain: ______________________________

  3. Two marbles drawn at the same time. Which case does it match? ______________ Explain:

_________________________________________________________________
  1. Error analysis. "Putting the marble back makes the draws dependent, because now you might draw the very same marble twice." What is wrong, and what is the correct classification?
_________________________________________________________________

PAGE 18 — Review Part B

Chapter 15 Review · Part B

Comparing and contrasting the two cases (8.PS.1b)

  1. Complete the comparison table.
Independent Dependent
Does draw 1 change draw 2?
Replacement
Second-stage denominator
Rule

  1. Bag of 3 red and 2 blue: P(both red)P(\text{both red}) with ____________ without ____________ Greater: ______

  2. Why is the first factor the same either way?

_________________________________________________________________
  1. Same bag: P(one of each color, either order)P(\text{one of each color, either order}) with ____________ without ____________ Greater: ______ Why?
_________________________________________________________________

  1. Bag of 5 red and 3 white: P(both white)P(\text{both white}) with ______ (≈\approx ______) without ______ (≈\approx ______)

  2. Reasoning. Why do the two answers get closer as the collection gets larger?

_________________________________________________________________

PAGE 19 — Review Parts C and D

Chapter 15 Review · Parts C and D

Two independent events (8.PS.1c)

  1. One flip and one roll: P(tails and 2)=P(\text{tails and } 2) = ____________

  2. Two dice: P(both 5)=P(\text{both } 5) = ____________

  3. 5-sector spinner spun twice: P(1 then 5)=P(1 \text{ then } 5) = ____________

  4. 4 green and 6 yellow, with replacement: P(yellow then green)=P(\text{yellow then green}) = ____________

  5. Two independent signals, each green 34\frac{3}{4} of the time: P(both green)=P(\text{both green}) = ____________

  6. Application. A player makes 45\frac{4}{5} of her free throws; two attempts, independent. P(both made)=P(\text{both made}) = ______ == ______%

Two dependent events (8.PS.1d)

  1. 5 red and 3 white, no replacement: P(both red)=P(\text{both red}) = ____________

  2. Two cards from 52, no replacement: P(both kings)=P(\text{both kings}) = ____________

  3. 7 blue and 3 red chips, no replacement: P(both red)=P(\text{both red}) = ____________

  4. 5 blue and 7 black pens, no replacement: P(both blue)=P(\text{both blue}) = ____________

  5. 6 black and 4 blue socks, no replacement: P(matching pair)=P(\text{matching pair}) = ____________

  6. Application. 20 raffle tickets, 4 winners; two drawn and kept. P(both win)=P(\text{both win}) = ____________

PAGE 20 — Review Part E

Chapter 15 Review · Part E

Mixed application and reasoning

FIGURE: fig9-blank-tree-templates.png (full width)

  1. Use the templates above for a bag of 3 red and 2 blue marbles. Fill Tree 1 for two draws with replacement and Tree 2 for two draws without replacement. Label every branch, compute all four products, and check that each total is 11.

  2. Application. A bowl holds 4 apples and 5 oranges. Two are taken and eaten. P(both apples)=P(\text{both apples}) = ____________

Work space:
  1. Application. Now the first piece is put back before the second is taken. P(both apples)=P(\text{both apples}) = ____________ Compare with item 127:
_________________________________________________________________
  1. Error analysis. For two cards dealt without replacement, a student writes P(both aces)=452⋅452=1169P(\text{both aces}) = \frac{4}{52} \cdot \frac{4}{52} = \frac{1}{169}. Explain the error and give the correct probability.
_________________________________________________________________
  1. Reasoning. Design your own bag. Write one problem about two independent events and one about two dependent events, solve both, and say which probability is larger and why.
Bag contents: _________________________________________________

Independent problem: __________________________________________

Solution: _____________________________________________________

Dependent problem: ___________________________________________

Solution: _____________________________________________________

Which is larger, and why? _____________________________________

PAGE 21 — Chapter self-check

Before You Move On

Check each box you can do without looking back.

Answer keys for every item are in Appendix A.