Grade 7 Workbook — Chapter 16: Probability: Theoretical and Experimental

SOL 7.PS.1 · Companion to Textbook Chapter 16

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/. Item numbers match the textbook exactly, and the problems are the same problems with the same numbers.

Materials for this chapter: one fair six-sided die, one coin, and a pencil. Several pages ask you to run real trials and record them.

PAGE 1 — Chapter opener

Chapter 16 · Probability: Theoretical and Experimental

Standard 7.PS.1

In this chapter you will:

• Determine the theoretical probability of an event
• Determine the experimental probability of an event from real trial data
• Describe how the experimental probability changes as the number of trials grows
• Compare probability found by experiment or simulation with theoretical probability

Words to know: probability experiment · outcome · sample space · event · favorable outcome · equally likely · theoretical probability · impossible event · certain event · complement · tree diagram · trial · frequency · tally · statistical investigation · experimental probability · independent · gambler's fallacy · expected frequency · variability · simulation

The two formulas for this chapter.

$$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \qquad P_{\text{exp}}(\text{event}) = \frac{\text{times it happened}}{\text{number of trials}}$$

Every probability is between $0$ and $1$. If you get a number outside that range, check your arithmetic.

PAGE 2 — Sample space and the probability scale

16.1 Theoretical Probability

FIGURE: fig5-die-event-sample-space.png (full width)

Fill in the blanks.

The complete list of every possible result is the ________________ .

One single possible result is an ________________ .

An outcome you are counting for your event is called a ________________ outcome.

FIGURE: fig1-probability-scale.png (full width)

Label each event on the scale by writing its letter above the right spot.

PAGE 3 — Spinners: read the angles

Size Is Everything

FIGURE: fig2-equal-spinner.png (half width) · FIGURE: fig3-unequal-spinner.png (half width)

Equal spinner (four sectors of $90°$).

$$P(A) = \underline{\hspace{2cm}} \qquad P(B) = \underline{\hspace{2cm}} \qquad P(C) = \underline{\hspace{2cm}} \qquad P(D) = \underline{\hspace{2cm}}$$

Unequal spinner. Fill in the table from the angles, not from the number of colors.

Watch out. Why is $P(\text{blue})$ on the second spinner not equal to $\tfrac14$?

PAGE 4 — Two coins, and guided practice

Listing a Sample Space

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

$$P(\text{two heads}) = \underline{\hspace{1.5cm}} \qquad P(\text{exactly one head}) = \underline{\hspace{1.5cm}} \qquad P(\text{at least one head}) = \underline{\hspace{1.5cm}}$$

Guided practice

1. List the sample space for one roll of a fair six-sided die. How many outcomes does it contain?

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2. For one roll of that die, find $P(3)$. ______

3. For one roll of that die, find $P(\text{odd})$ in lowest terms. ______

4. On the spinner with four equal $90°$ sectors A, B, C, D, find $P(B)$. ______

5. A bag holds 4 red, 3 blue, and 5 green marbles. One is drawn without looking. Find $P(\text{red})$. ______

6. Using item 5 and the complement, find $P(\text{not red})$. ______

PAGE 5 — Independent practice 16.1

Practice · Theoretical Probability

7. A fair six-sided die is rolled once. Find each in lowest terms.

   a) $P(5)$ ______ b) $P(\text{greater than }4)$ ______ c) $P(7)$ ______ d) $P(\text{less than }7)$ ______

8. A fair coin is flipped twice. List the four outcomes, then find $P(\text{exactly one head})$ and $P(\text{two tails})$.

   Sample space: _______________________________________________

   $P(\text{exactly one head}) =$ ______ $P(\text{two tails}) =$ ______

9. On the unequal spinner (red $180°$, blue $90°$, green $45°$, yellow $45°$), find $P(\text{red})$ ______, $P(\text{blue})$ ______, $P(\text{green or yellow})$ ______. Show that all four sector probabilities add to 1.

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10. From the bag of 4 red, 3 blue, and 5 green marbles, find $P(\text{green})$ and $P(\text{not green})$.

    	Fraction	Decimal (thousandth)	Percent (tenth)
    $P(\text{green})$
    $P(\text{not green})$

11. Eleven tiles show the letters of PROBABILITY, one per tile. One is drawn. $P(\text{B}) =$ ______ $P(\text{vowel}) =$ ______

12. A spinner has eight equal sectors numbered 1–8. $P(\text{a multiple of }3) =$ ______

13. For one roll of a fair die, describe one event with probability $0$ and one with probability $1$.

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14. Order these four events for one die roll from least likely to most likely: rolling a 2; rolling a number greater than 0; rolling a number greater than 2; rolling a negative number.

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15. Application. A raffle sells 25 tickets and draws one winner. Nadia buys 4. Find the probability she wins as a fraction ______, a decimal ______, and a percent ______.

16. Reasoning. Why must the probabilities of all outcomes in a sample space add to exactly 1? Use the unequal spinner in your explanation.

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PAGE 6 — Exit ticket 16.1

Exit Ticket · Lesson 16.1

Name: ________________________ Date: ____________

17. A fair six-sided die is rolled once. Find $P(2)$. ______

18. On the spinner with four equal sectors A, B, C, D, find $P(A \text{ or } C)$. ______

19. From the bag of 4 red, 3 blue, and 5 green marbles, find $P(\text{blue})$ in lowest terms. ______

20. Explain the difference between an event with probability $0$ and one with probability $1$. Give an example of each for a single die roll.

PAGE 7 — Hands-on: roll a die 60 times

16.2 Experimental Probability · Investigation 1

You need one fair six-sided die. Roll it 60 times. Make one tally mark for each roll. Then count.

Check first. Add your six frequencies. If the total is not exactly 60, recount before going on.

Now compute your own experimental probabilities.

$$P_{\text{exp}}(\text{even}) = \frac{\underline{\hspace{1cm}}}{60} = \underline{\hspace{2cm}} \qquad P_{\text{exp}}(\text{greater than }4) = \frac{\underline{\hspace{1cm}}}{60} = \underline{\hspace{2cm}}$$

PAGE 8 — Hands-on: flip a coin 40 times

Investigation 2 · Forty Flips

You need one coin. Flip it 40 times, tallying as you go.

$$P_{\text{exp}}(\text{heads}) = \frac{\underline{\hspace{1cm}}}{40} = \underline{\hspace{2cm}} \qquad P_{\text{exp}}(\text{tails}) = \frac{\underline{\hspace{1cm}}}{40} = \underline{\hspace{2cm}}$$

Add your two experimental probabilities. Sum: ______ Why must it equal 1?

Compare with a classmate. Their $P_{\text{exp}}(\text{heads})$: ______ Yours: ______

Did you get the same number? Can both records be correct? Explain.

PAGE 9 — The class data, and guided practice

Reading Someone Else's Investigation

A class rolled one fair six-sided die 60 times. Use this record for items 21–24 and for the rest of the chapter.

A student flipped a fair coin 40 times: 23 heads, 17 tails.

A student spun the four-equal-sector spinner 80 times: A: 24, B: 17, C: 21, D: 18.

Guided practice

21. Add the six frequencies in the 60-roll table and confirm the total is 60. Why must you do this first?

    Total: ______ Reason: _______________________________________________

22. $P_{\text{exp}}(1)$ in lowest terms: ______

23. $P_{\text{exp}}(6)$ in lowest terms: ______

24. $P_{\text{exp}}(\text{greater than }4)$: ______

25. From the 40 flips (23 heads, 17 tails), find $P_{\text{exp}}(\text{tails})$. ______

26. From the 80 spins, find $P_{\text{exp}}(C)$. ______

PAGE 10 — Independent practice 16.2

Practice · Experimental Probability

27. Using the 60-roll table, complete the table.

    Outcome	2	4	5
    Fraction, lowest terms
    Decimal (hundredth)

28. Using the 60-roll table, find $P_{\text{exp}}(\text{odd})$ ______. Show that $P_{\text{exp}}(\text{odd}) + P_{\text{exp}}(\text{even}) = 1$ and explain why that had to happen.

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29. From the 80 spins (A: 24, B: 17, C: 21, D: 18), find $P_{\text{exp}}(B)$ ______, $P_{\text{exp}}(D)$ ______, $P_{\text{exp}}(B \text{ or } D)$ ______

30. A spinner was spun 50 times: A: 14, B: 11, C: 13, and D smudged.

    Sector	A	B	C	D	Total
    Frequency	14	11	13		50

    Missing frequency for D: ______ $P_{\text{exp}}(D) =$ ______

31. A thumbtack was tossed 60 times and landed point up 33 times. $P_{\text{exp}}(\text{point up}) =$ ______ $P_{\text{exp}}(\text{point down}) =$ ______

    Why can't theoretical probability be used here? _______________________________________________

32. A player made 12 of 20 free throws. $P_{\text{exp}}(\text{make}) =$ ______ Estimated makes in 100 attempts: ______

33. Application. An arcade game was played 250 times and gave a prize 30 times. $P_{\text{exp}}(\text{prize}) =$ ______ (lowest terms) $=$ ______ %

34. Application. An inspector checked 400 phone cases and found 16 scratched. $P_{\text{exp}}(\text{scratched}) =$ ______ Expected scratched in a shipment of 2{,}500: ______

35. Reasoning. Why can't you determine an experimental probability just by thinking carefully about the situation?

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36. Reasoning. Two students flip the same fair coin 20 times. One gets 9 heads, the other 13. Compute both experimental probabilities and explain how both can be correct.

    ______ and ______ _______________________________________________

PAGE 11 — Exit ticket 16.2

Exit Ticket · Lesson 16.2

Name: ________________________ Date: ____________

37. A spinner was spun 30 times and landed red 12 times. $P_{\text{exp}}(\text{red}) =$ ______

38. Using the 60-roll class table, $P_{\text{exp}}(5) =$ ______

39. A coin flipped 40 times gave 23 heads. $P_{\text{exp}}(\text{heads}) =$ ______ as a decimal

40. If the class rolled the same die 60 more times, would the frequency table come out identical? What would change, and what would stay the same?

PAGE 12 — One thousand flips

16.3 What Happens as Trials Grow

One fair coin, simulated 1{,}000 times. Each row counts every flip made so far.

FIGURE: fig6-convergence-coin.png (full width)

Circle the correct wording.

As trials increase, the experimental probability ( always equals / tends to get closer to ) the theoretical probability.

After six heads in a row, the next flip is ( more likely tails / still $\tfrac12$ heads ).

The key sentence. Copy it.

Each trial is independent. The coin has no memory, so no outcome is ever "due."

PAGE 13 — Hands-on: watch it settle

Investigation 3 · Your Own Running Record

Flip a coin 50 times. Tally in blocks of 10, then fill in the running totals.

Did the distance from $0.5$ shrink at every step? ______

If not, does that mean something went wrong? Explain.

Combine with the whole class. Class total flips: ______ Class total heads: ______

Class $P_{\text{exp}}(\text{heads})$: ______ Is it closer to $0.5$ than your own 50-flip result? ______

Is it guaranteed to be closer? ______ Explain: _______________________________________________

PAGE 14 — Practice 16.3

Practice · Growing the Number of Trials

Use the 1{,}000-flip table on page 12.

Guided practice

41. $P_{\text{exp}}(\text{heads})$ after 100 flips: ______ (lowest terms) ______ (decimal)

42. Distance from $0.5$ at 20 flips ______ and at 500 flips ______. Closer: ______

43. Did the distance shrink at every step? ______ Name the step where it grew: ______

44. A fair coin lands heads six times in a row. $P(\text{next flip is heads}) =$ ______

45. Distance of the head count from exactly half the flips: at 100 flips ______, at 1{,}000 flips ______

46. As the number of trials increases, the experimental probability ____________________ the theoretical probability.

Independent practice

47. The 500-flip row: fraction in lowest terms ______, decimal ______

48. Order the 20-, 100-, and 1{,}000-flip rows from farthest to closest to $0.5$, showing all three distances.

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49. Owen says the experimental probability after 1{,}000 flips will be exactly $0.5$. Use the table to show he is wrong, then rewrite his statement correctly.

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50. Priya spun blue five times in a row and says blue is now less likely. What is wrong with her reasoning? Use the word independent.

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51. A die rolled 60 times gave a 6 fifteen times. $P_{\text{exp}}(6) =$ ______ $=$ ______ (decimal). After 600 rolls, what would you expect, and how confident should you be?

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52. Which do you trust more, an estimate from 30 trials or from 3{,}000 trials? Is the one you chose guaranteed to be closer to the true value?

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53. Application. Short test: 20 plays, 8 wins. Long test: 200 plays, 46 wins. $P_{\text{exp}}$ short ______ $P_{\text{exp}}$ long ______ Better estimate: ______

54. Application. A die rolled 600 times gave a 6 ninety-seven times. $P_{\text{exp}}(6) \approx$ ______ Compare to $\tfrac16$. Unfair?

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55. Reasoning. What does "the coin has no memory" mean, and why doesn't it contradict the drift toward $0.5$?

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56. Reasoning. How can the gap between the head count and half the flips grow while the experimental probability gets closer to $0.5$?

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PAGE 15 — Exit ticket 16.3

Exit Ticket · Lesson 16.3

Name: ________________________ Date: ____________

57. Which row of the 1{,}000-flip table is closest to $0.5$? Row: ______ Distance: ______

58. A fair coin lands tails four times in a row. $P(\text{next flip is tails}) =$ ______

59. True or false: rolling a die enough times guarantees $P_{\text{exp}}(6)$ will equal $\tfrac16$ exactly. ______

    Explain: _______________________________________________

60. In one careful sentence, describe what happens to the experimental probability as trials increase.

PAGE 16 — Side by side

16.4 Comparing the Two Probabilities

$$\text{expected frequency} = P(\text{event}) \times \text{number of trials}$$

Complete the comparison for the 60-roll class data.

FIGURE: fig7-theoretical-vs-experimental-die.png (full width)

Guided practice

61. $P(3) =$ ______ $P_{\text{exp}}(3) =$ ______ Difference (thousandth): ______

62. Expected frequency of a 5 in 60 rolls: ______

63. The class rolled twelve 2s. Difference from the expected frequency: ______

64. $P(\text{even}) =$ ______ $P_{\text{exp}}(\text{even}) =$ ______

65. Expected heads in 40 flips: ______ Difference from the actual 23: ______

66. Expected frequency for sector A in 80 spins: ______ Difference from Ana's 24: ______

PAGE 17 — Practice 16.4

Practice · Theoretical vs. Experimental

67. Complete the comparison table for the 60-roll class data.

    Outcome	1	3	4
    Frequency out of 60
    $P_{\text{exp}}$ as a fraction
    $P_{\text{exp}}$ as a decimal (thousandth)
    Difference from $0.167$

68. Compare $P(\text{greater than }4)$ with $P_{\text{exp}}(\text{greater than }4)$. ______ and ______

    What does an exact match tell you — and not tell you? _______________________________________________

69. The 500-flip row had 255 heads. Expected frequency: ______ Difference: ______

70. A bag of 4 red, 3 blue, 5 green marbles is drawn from and replaced 120 times. Expected frequency of green: ______

71. Describe how to simulate 100 spins of the four-equal-sector spinner with a random number generator. Say exactly which numbers stand for which sectors.

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72. A die rolled 60 times gave a 6 thirty-four times. $P_{\text{exp}}(6) =$ ______ $\approx$ ______ Likely fair? ______ Why?

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73. Devon says any difference between experimental and theoretical probability means someone counted wrong. What is Devon missing?

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74. Application. A game advertises that $\tfrac15$ of plays win. A club plays 300 times and wins 41. Expected wins: ______ $P_{\text{exp}}(\text{win}) \approx$ ______ Is the advertisement accurate?

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75. Application. A company says $\tfrac16$ of boxes hold a prize. A store opens 90 boxes and finds 14 prizes. Expected: ______ $P_{\text{exp}} \approx$ ______ Consistent with the claim? ______

76. Reasoning. Give two reasons experimental and theoretical probability can differ.

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77. Reasoning. How would you decide whether a difference is ordinary variability or evidence the model is wrong?

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78. Reasoning. Can an event with theoretical probability greater than 0 have experimental probability exactly 0? Give a numerical example.

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PAGE 18 — Exit ticket 16.4

Exit Ticket · Lesson 16.4

Name: ________________________ Date: ____________

79. $P(1)$ for a fair die: ______ $P_{\text{exp}}(1)$ from the 60-roll table: ______

80. Expected heads in 200 flips: ______ The table recorded 96. Difference: ______

81. A four-equal-sector spinner spun 40 times lands on A ten times. $P(A) =$ ______ $P_{\text{exp}}(A) =$ ______

82. In two or three sentences, explain the difference between theoretical and experimental probability, and name a situation where only the experimental one can be found.

PAGE 19 — Chapter 16 review, part 1

Chapter 16 Review

Part A · Theoretical probability

83. Fair die: $P(3) =$ ______ $P(\text{odd}) =$ ______ $P(\text{less than }3) =$ ______

84. Bag of 4 red, 3 blue, 5 green: $P(\text{red}) =$ ______ $P(\text{blue}) =$ ______ $P(\text{not green}) =$ ______

85. Unequal spinner: $P(\text{red}) =$ ______ $P(\text{green or yellow}) =$ ______

86. Two coin flips. Sample space: _______________________

    $P(\text{two heads}) =$ ______ $P(\text{at least one head}) =$ ______

87. One die roll: an event with probability $0$: ______________ with probability $1$: ______________

88. Spinner with eight equal sectors 1–8: $P(\text{greater than }5) =$ ______

89. If $P(E) = \tfrac27$, then $P(\text{not }E) =$ ______

Part B · Experimental probability

A fair six-sided die was rolled 100 times. Use this record for items 90–94.

90. $P_{\text{exp}}(2) =$ ______ (fraction) ______ (decimal)

91. $P_{\text{exp}}(6) =$ ______ (lowest terms)

92. $P_{\text{exp}}(\text{even}) =$ ______ (lowest terms)

93. $P_{\text{exp}}(\text{odd}) =$ ______ Show it plus $P_{\text{exp}}(\text{even})$ equals 1: ______________

94. $P_{\text{exp}}(\text{less than }3) =$ ______

95. A coin flipped 80 times came up heads 37 times. $P_{\text{exp}}(\text{heads}) =$ ______ $P_{\text{exp}}(\text{tails}) =$ ______

PAGE 20 — Chapter 16 review, part 2

Chapter 16 Review (continued)

Part C · As the number of trials increases

96. From the 1{,}000-flip table: distance from $0.5$ at 20 flips ______, at 1{,}000 flips ______

97. Better estimate — the 10-flip result or the 1{,}000-flip result? ______

    Explain, accounting for the 10-flip result being exactly $0.5$: _______________________________________________

98. A fair coin lands heads seven times in a row. $P(\text{next flip is heads}) =$ ______

99. True or false: after enough flips the experimental probability becomes exactly $0.5$. ______ Explain: ______________

100. Write one careful sentence about what happens to the experimental probability as trials increase.

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Part D · Comparing experimental and theoretical

101. Using the 100-roll table: $P(3) =$ ______ $P_{\text{exp}}(3) =$ ______ Difference (thousandth): ______

102. Expected frequency of each outcome in 100 rolls (nearest tenth): ______

     Outcome whose actual frequency is farthest from it: ______

103. Expected heads in 80 flips: ______ Compared with the 37 in item 95: ______

104. A die rolled 120 times gave a 6 forty-two times. Expected: ______ $P_{\text{exp}}(6) =$ ______ Likely fair? ______

105. How would you simulate 100 die rolls with random digits? What do you do with 7, 8, 9, and 0?

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PAGE 21 — Chapter 16 review, part 3

Chapter 16 Review (continued)

Part E · Mixed application and reasoning

106. Application. A raffle sells 250 tickets and draws one winner. Trey buys 10.

     $P(\text{Trey wins}) =$ ______ $=$ ______ % $P(\text{Trey does not win}) =$ ______ $=$ ______ %

107. Application. The unequal spinner (red $\tfrac12$, blue $\tfrac14$, green $\tfrac18$, yellow $\tfrac18$) was spun 400 times.

     Color	Red	Blue	Green	Yellow	Total
     Frequency	191	108	45	56
     Expected frequency					400
     Difference					0

     Do the frequencies total 400? ______ $P_{\text{exp}}(\text{red}) =$ ______ (decimal)

     Consistent with the stated sector sizes? _______________________________________________

108. Reasoning. Why can no probability equal $1.4$ or $-0.2$?

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109. Reasoning. After four reds in a row on that spinner, a player says red is now less likely. Explain the error using the word independent, and give the correct probability for the next spin. ______

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110. Reasoning. Explain the difference between theoretical and experimental probability, and describe a situation where only experimental probability can be determined.

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Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Tally tables (pages 7, 8, 13): make the tally column at least 2.5 in wide and the row height at least 0.4 in so students have room for five-bar gates; these pages are used with a real die and coin
• Figure widths: fig1-probability-scale.png, fig4-two-coin-sample-space.png, fig5-die-event-sample-space.png, fig6-convergence-coin.png, and fig7-theoretical-vs-experimental-die.png are landscape and run full width; fig2-equal-spinner.png is square and fig3-unequal-spinner.png is slightly wide — place them side by side on page 3 at half width each
• Page 12: the table and fig6-convergence-coin.png must stay on the same page so students can match rows to points on the curve
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow
• Materials callout: repeat the die-and-coin materials note on pages 7, 8, and 13