Represent Sample Spaces for Compound Events - 7th Grade Math

1. What does a table show for a compound event sample space?

   Outcomes arranged by rows and columns for two actions. Each row represents one action's choices, columns show the other's.

2. What is the difference between an outcome and an event?

   Outcome: one result; Event: a set of outcomes. An outcome is singular; an event contains one or more outcomes.

3. What does an ordered pair outcome like $(H,3)$ mean in a compound event?

   First result is $H$, second result is $3$; order matters. The first element shows the first event's result, the second shows the second event's result.

4. Which representation is most appropriate to show outcomes for $3$ sequential choices: list, table, or tree diagram?

   Tree diagram. Trees show branching paths for sequential decisions.

5. Identify the number of outcomes for flipping three coins.

   $8$. Each coin has 2 outcomes: $2 \times 2 \times 2 = 8$ total.

6. What is the sample space for rolling a die once, written as an organized list?

   $\{1,2,3,4,5,6\}$. Standard die shows numbers $1$ through $6$.

7. Identify the outcomes for: flip two coins; event is "exactly one head" (use ordered pairs).

   $\{(H,T),(T,H)\}$. One $H$ and one $T$ in either order.

8. Identify the outcomes for choosing a letter from $\{A,B\}$ then a number from $\{1,2,3\}$.

   $(A,1),(A,2),(A,3),(B,1),(B,2),(B,3)$. Each letter pairs with each number: $2 \times 3 = 6$ total.

9. Which word indicates an intersection of conditions in everyday language: "and" or "or"?

   "and". Both conditions must be true simultaneously.

10. What is the sample space size for flipping a coin, then choosing a number from $\{1,2,3\}$?

    $6$ outcomes. Coin has $2$ outcomes, numbers have $3$, so $2 \times 3 = 6$.

11. Identify the event outcomes for "at least one $6$" when rolling two dice $(a,b)$.

    $\{(6,1),(6,2),(6,3),(6,4),(6,5),(6,6),(1,6),(2,6),(3,6),(4,6),(5,6)\}$. Includes all pairs where first or second die shows $6$.

12. What is the number of outcomes when flipping a coin twice?

    $4$. Two flips with two outcomes each gives $2 \times 2 = 4$.

13. Identify the outcomes for: roll a die; event is "number less than $3$".

    $\{1,2\}$. Only values 1 and 2 are less than 3.

14. What is the number of outcomes when rolling two standard dice (ordered pairs)?

    $36$. Six faces on first die times six on second equals $6 \times 6$.

15. Identify the event outcomes for "both dice show the same number" when rolling two dice $(a,b)$.

    $\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\}$. Doubles occur when both dice show the same value.

16. Identify the sample space for flipping a coin and rolling a die as ordered pairs $(\text{coin},\text{die})$.

    $(H,1)$ to $(H,6)$ and $(T,1)$ to $(T,6)$. Lists all $12$ combinations: $2$ coin outcomes times $6$ die outcomes.

17. What does a tree diagram show for a compound event sample space?

    Branching choices for each step, ending in all possible outcomes. Each branch represents a choice, paths show combined outcomes.

18. Identify the event outcomes for "sum is $7$" when rolling two standard dice $(a,b)$.

    $\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$. All pairs where first die plus second die equals $7$.

19. Identify the event outcomes for "rolling a number greater than $4$" on one die.

    $\{5,6\}$. Only $5$ and $6$ are greater than $4$ on a standard die.

20. What does an organized list show for a compound event?

    Every possible outcome written in a systematic order. Lists help ensure no outcomes are missed or repeated.

21. What is the sample space size for tossing a fair coin $3$ times?

    $8$ outcomes. Each flip has $2$ outcomes, so $2^3 = 8$ total.

22. What is the counting rule for total outcomes when one event has $m$ outcomes and another has $n$ outcomes?

    Total outcomes $=m\times n$. Fundamental counting principle for independent events.

23. Identify the number of outcomes in the sample space for choosing 1 of $4$ shirts and 1 of $3$ pants.

    $12$ outcomes. Multiply choices: $4$ shirts $\times$ $3$ pants $= 12$.

24. Which outcomes describe "exactly one head" when flipping two coins as $(\text{first},\text{second})$?

    $(H,T),(T,H)$. One $H$ and one $T$ in either order.

25. What is the event (as outcomes) for "sum is $7$" when rolling two dice as $(d_1,d_2)$?

    $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$. All pairs of dice values that add to exactly $7$.

26. Identify the sample space when rolling a die and flipping a coin (die first).

    $(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)$. Each pair shows (die result, coin result) for all combinations.

27. Identify the event outcomes for "at least one head" when a coin is tossed twice.

    $\{HH,HT,TH\}$. All outcomes except $TT$ have at least one head.

28. Identify the event outcomes for "at least one head" in two coin flips.

    ${HH,HT,TH}$. All outcomes except TT contain at least one head.

29. Identify the event outcomes for "exactly one head" in two coin flips.

    ${HT,TH}$. Only HT and TH have one H each; HH has two, TT has none.

30. Which representation is most appropriate for outcomes of two categories (rows and columns): list, table, or tree diagram?

    Table. Tables organize two-dimensional data in rows and columns.

31. Identify the outcomes for the event "first is odd, second is even" when rolling two dice as $(d_1,d_2)$.

    $\{(1,2),(1,4),(1,6),\dots,(5,6)\}$ ($9$ outcomes). Odd first die $(1,3,5)$ times even second die $(2,4,6)$: $3 \times 3 = 9$.

32. What is the sample space size for rolling one standard die and flipping one coin?

    $12$ outcomes. Die has $6$ outcomes, coin has $2$, so $6 \times 2 = 12$.

33. Identify the number of outcomes in the event "sum is $2$ or $12$" when rolling two dice.

    $2$ outcomes. Only $(1,1)$ sums to $2$ and $(6,6)$ sums to $12$.

34. What is the sample space size for rolling two standard dice, using ordered pairs $(a,b)$?

    $36$ outcomes. Each die has $6$ outcomes, so $6 \times 6 = 36$ pairs.

35. Identify the outcomes for "at least one $6$" when rolling two dice (ordered pairs).

    All $(6,k)$ or $(k,6)$ for $k=1$ to $6$, counting $(6,6)$ once. Includes all pairs where first is 6 OR second is 6 (or both).

36. What are the outcomes for the event "flip a coin and get heads, then roll an even number"?

    $(H,2),(H,4),(H,6)$. Heads pairs with each even number: $2$, $4$, and $6$.

37. What is the organized-list sample space for flipping two coins (order: first, second)?

    $\{(H,H),(H,T),(T,H),(T,T)\}$. All 4 combinations of H/T for first coin and H/T for second coin.

38. Which option correctly represents "or" in an event description?

    Outcomes that satisfy at least one condition (union). Either condition (or both) being true makes the outcome count.

39. Identify the outcomes for "first is odd and second is even" when rolling two dice.

    All $(a,b)$ with $a\in\{1,3,5\}$ and $b\in\{2,4,6\}$. First die shows 1, 3, or 5; second shows 2, 4, or 6.

40. Identify the sample space for flipping a coin then rolling a die (use ordered pairs).

    $\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)\}$. Lists all 12 combinations: 2 coin outcomes × 6 die outcomes.

41. Identify the event outcomes for "exactly one head" when a coin is tossed twice.

    $\{HT,TH\}$. Only $HT$ and $TH$ have exactly one head.

42. Which representation is best for two events with two categories each (like $H/T$ and $R/B$): list, table, or tree?

    Table. Creates a grid showing all combinations of two binary events efficiently.

43. What is the sample space of a compound event?

    The set of all possible outcomes of the combined actions. Includes every result that could happen when multiple actions occur together.

44. What is the sample space of a compound event?

    The set of all possible outcomes of the combined actions. Includes every combination that could happen when multiple events occur together.

45. Identify the number of outcomes when rolling two standard number cubes.

    $36$. Each die has 6 outcomes: $6 \times 6 = 36$ total combinations.

46. What is an outcome in a sample space?

    One specific result of an experiment (one element of the sample space). A single possibility from the complete list of what could happen.

47. Identify the event outcomes for "no heads" in two coin flips.

    ${TT}$. Only TT has zero heads; all others have at least one.

48. Identify the outcomes for: flip a coin then roll a die; event is "tails and an even number".

    $\{(T,2),(T,4),(T,6)\}$. Only pairs where first is $T$ and second is 2, 4, or 6.

49. What is the sample space (ordered) for spinning $\{1,2\}$ and flipping a coin $\{H,T\}$ as $(\text{spin},\text{flip})$?

    $(1,H),(1,T),(2,H),(2,T)$. Each spin result pairs with each flip result: $2 \times 2 = 4$.

50. What is the number of outcomes when rolling a die and flipping a coin?

    $12$. Six die faces times two coin sides equals $6 \times 2 = 12$.

51. What is the sample space for a compound event?

    The set of all possible outcomes of the combined actions. Includes every combination from all steps of the compound event.

52. Identify the outcomes for: roll two dice; event is "sum is $7$" (use ordered pairs).

    $\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$. All pairs of dice values that add to exactly 7.

53. Identify the sample space when flipping a coin twice (use ordered outcomes).

    ${HH,HT,TH,TT}$. First letter shows first flip, second shows second flip result.

54. Identify the number of outcomes when flipping a coin and rolling a die.

    $12$. Multiply: 2 coin outcomes × 6 die outcomes = 12 total.

55. What multiplication rule finds the number of outcomes for two independent steps with $m$ and $n$ outcomes?

    Total outcomes $= m \times n$. Multiply the number of outcomes at each independent step.

56. Identify the outcomes for "different colors" when drawing one sock from $\{R,B\}$ then another from $\{R,B\}$ with order.

    $(R,B)$ and $(B,R)$. Red-blue and blue-red are the only non-matching pairs.

57. Identify the event outcomes for "rolling an even number" on one die.

    $\{2,4,6\}$. Even numbers on a die are $2$, $4$, and $6$.

58. Which representation is best for two-step events with many branches: organized list, table, or tree diagram?

    Tree diagram. Shows branching paths clearly when each step has multiple possible outcomes.

59. Which outcomes describe "at least one head" when flipping two coins as $(\text{first},\text{second})$?

    $(H,H),(H,T),(T,H)$. Excludes only $(T,T)$ - all other outcomes have at least one $H$.

60. Identify the outcomes for "both even" when rolling one die then another (ordered).

    $(2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6)$. Both dice must show 2, 4, or 6 (the three even numbers).

61. Identify the outcomes for "sum is $5$" when rolling two dice (ordered pairs).

    $(1,4),(2,3),(3,2),(4,1)$. These are the only pairs where first die + second die = 5.

62. Identify the event outcomes for "no heads" when a coin is tossed twice.

    $\{TT\}$. Only $TT$ has zero heads.

63. What is the sample space for tossing a coin twice, written as an organized list?

    $\{HH,HT,TH,TT\}$. Lists all possible ordered outcomes when flipping twice.

64. Identify the outcomes for: roll two dice; event is "at least one die shows $6$" (ordered pairs).

    $\{(6,1),(6,2),(6,3),(6,4),(6,5),(6,6),(1,6),(2,6),(3,6),(4,6),(5,6)\}$. Includes all pairs where first die is 6 OR second die is 6.

65. What does an ordered pair like $(H,3)$ represent in a sample space?

    A specific outcome in order: first $H$, then $3$. Shows the exact sequence: first outcome followed by second outcome.

66. Identify the outcomes for: choose a letter from $\{A,B\}$ then a number from $\{1,2,3\}$.

    $\{(A,1),(A,2),(A,3),(B,1),(B,2),(B,3)\}$. All 6 ordered pairs: 2 letters × 3 numbers.

67. What is the sample space size for drawing 1 of 3 shirts and 1 of 4 pants (order fixed: shirt, pants)?

    $12$. Multiply: 3 shirt choices × 4 pants choices = 12 outfits.

68. Which option correctly represents "and" in an event description?

    Outcomes that satisfy both conditions at the same time (intersection). Both conditions must be true for the outcome to count.

69. What are the outcomes for the event "roll a die and get a number greater than $4$, then flip tails"?

    $(5,T),(6,T)$. Numbers greater than $4$ are $5$ and $6$, each paired with tails.

70. Identify the outcomes for: roll a die; event is "number less than $3$ or a $6$".

    $\{1,2,6\}$. Combines outcomes less than 3 (1,2) with the outcome 6.

71. What is the difference between $(H,T)$ and $(T,H)$ in a compound event?

    They are different outcomes because order matters in ordered pairs. First position shows first action, second shows second action.

72. Which word indicates a union of conditions in everyday language: "and" or "or"?

    "or". At least one condition must be true.

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