Skip to main content
Sign in Create a free account
Varsity Tutors
Sign in

Represent Sample Spaces for Compound Events - 7th Grade Math

  1. A carnival game involves spinning two wheels sequentially. Wheel 1 has sections numbered 1, 2, 3, and Wheel 2 has sections colored Red, Blue, Yellow, Green. The tree diagram represents all possible outcomes. Based on this representation, if a player wins when they get an odd number on Wheel 1 AND a primary color (Red, Blue, or Yellow) on Wheel 2, how many winning outcomes exist in the complete sample space?
    Answer: 6 winning outcomes out of 12 total outcomes in the sample space Sample space = 3 numbers × 4 colors = 12 total outcomes. Odd numbers are 1 and 3. Primary colors are Red, Blue, Yellow. Winning outcomes: (1,Red), (1,Blue), (1,Yellow), (3,Red), (3,Blue), (3,Yellow) = 6 outcomes. Choice A overcounts winning outcomes. Choice C incorrectly calculates total outcomes as 16. Choice D undercounts both winning outcomes and total outcomes.
  2. Miguel is planning outfits by choosing one shirt, one pair of pants, and one pair of shoes. He creates a table to organize all possibilities. Using the systematic table shown, if Miguel randomly selects an outfit, what is the total number of possible outfits, and how many outfits include both blue pants and sneakers?
    Answer: 24 total outfits; 4 outfits include both blue pants and sneakers Total outfits = 4 shirts × 3 pants × 2 shoes = 24. For outfits with blue pants AND sneakers: any of the 4 shirts can be chosen with blue pants and sneakers, giving 4 × 1 × 1 = 4 outfits. Choice B undercounts total outfits. Choice C overcounts blue pants + sneakers combinations. Choice D undercounts total outfits.
  3. A student flips a coin and then rolls a standard six-sided number cube (1-6). The event is: "heads and an even number." Which set lists all outcomes in this event?
    Answer: {H2,H4,H6}\{H2,H4,H6\} This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possibilities, such as for two coins {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches, such as for flipping twice yielding 4 paths. To identify events, read the description like 'both heads' as HH or 'at least one' as HH, HT, TH, then locate and count them. For example, a coin and die list is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for flipping twice shows HH, HT, TH, TT. The correct representation is choice B, listing only heads with even numbers {H2, H4, H6}. Common errors include listing odds in A, including tails in C, or all heads in D without checking even. When creating a list, systematically combine all possibilities; for tables, use rows and columns; for trees, branch out with paths as outcomes. To identify events, translate the language, locate matching outcomes, and count them, avoiding mistakes like incomplete sets, wrong event language, confusing tree branches, or miscounting cells.
  4. A board game uses two spinners: Spinner A with 3 equal sections (Win, Lose, Draw) and Spinner B with 2 equal sections (Bonus, No Bonus). Players spin both spinners at the same time. A tree diagram shows all possible outcomes. How many of the tree diagram's outcomes (final endpoints) satisfy the event 'Win or Draw on Spinner A AND Bonus on Spinner B', out of the total number of endpoints in the tree?
    Answer: 2 branches out of 6 total endpoints represent this event in the complete tree diagram The tree diagram has 3 x 2 = 6 total endpoints, one for each combination of a Spinner A result and a Spinner B result. The event 'Win or Draw on Spinner A AND Bonus on Spinner B' matches exactly two of these combinations: Win-Bonus and Draw-Bonus. So 2 out of the 6 total endpoints satisfy the event. Choice A overcounts by including outcomes that don't require Bonus. Choices C and D both incorrectly state 5 total endpoints, when the correct total is 6.
  5. Marcus spins a spinner with sections labeled Red, Blue, and Green, then rolls a standard six-sided die. He wins if he gets Blue on the spinner AND an even number on the die. Using the tree diagram shown, how many outcomes are in the sample space, and how many of those outcomes result in Marcus winning?
    Answer: Sample space has 18 outcomes; Marcus wins with 3 outcomes The sample space consists of all possible combinations of spinner outcomes (3) and die outcomes (6), giving 3 × 6 = 18 total outcomes. Marcus wins only when he gets Blue AND an even number (2, 4, or 6). This gives exactly 3 winning outcomes: (Blue, 2), (Blue, 4), and (Blue, 6).
  6. Two different number cubes labeled 1-6 are rolled. Event FF is: "Both numbers are odd." How many outcomes are in FF?
    Answer: 9 This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possible outcomes, such as for two coins {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches, such as for flipping a coin twice yielding 4 paths. For example, a coin flip followed by a die roll has a list {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and the event 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for two coin flips shows paths HH, HT, TH, TT. The correct count here is choice C, 9 outcomes where both are odd (3 odds per die: 3x3=9). Common errors include counting only one die's odds in A or overcounting in D. When creating a table, mark cells where both a and b are odd and count them. To identify and count, translate 'both odd' to qualifying pairs; mistakes often involve confusing 'both' with 'at least one' or wrong odd numbers.
  7. Coach Rivera organizes team selections by having students draw one ball from Box 1 (containing balls labeled X, Y, Z) and one ball from Box 2 (containing balls labeled 1, 2, 3, 4). Students are assigned to the advanced team if they draw a ball from the second half of the alphabet (Y or Z) from Box 1 AND an even number from Box 2. Using a systematic table to represent this sample space, what portion of all possible outcomes results in advanced team assignment?
    Answer: 4 favorable outcomes out of 12 total possible drawing combinations When you encounter probability questions involving two separate events, you need to systematically list all possible combinations to find the sample space and identify favorable outcomes. Let's create a table showing all possible combinations. Box 1 has letters {X, Y, Z} and Box 2 has numbers {1, 2, 3, 4}:
    Box 1Box 2Outcome
    X1(X,1)
    X2(X,2)
    X3(X,3)
    X4(X,4)
    Y1(Y,1)
    Y2(Y,2)
    Y3(Y,3)
    Y4(Y,4)
    Z1(Z,1)
    Z2(Z,2)
    Z3(Z,3)
    Z4(Z,4)
    This gives us 12 total possible outcomes (3 letters × 4 numbers = 12). For advanced team assignment, students need: (1) second half of alphabet letter (Y or Z) AND (2) even number (2 or 4). The favorable outcomes are: (Y,2), (Y,4), (Z,2), (Z,4) — that's 4 favorable outcomes. Choice A incorrectly counts 10 total outcomes, missing some combinations. Choice B correctly identifies 12 total outcomes but miscounts 6 favorable outcomes, likely including all Y and Z combinations regardless of the number requirement. Choice C suggests 14 total outcomes, which is mathematically impossible with only 3 letters and 4 numbers. Remember: when dealing with compound probability events, always create a systematic table or tree diagram to avoid missing combinations or miscounting requirements.
  8. A student randomly chooses one shirt (Red or Blue) and then one pair of shorts (Black, Gray, or Tan). How many outcomes are in the sample space?
    Answer: 6 This problem asks for the total number of outcomes when choosing one shirt from 2 options and one pair of shorts from 3 options. Using the counting principle, multiply the number of shirt choices by the number of shorts choices: 2 x 3 = 6, matching Choice D. Choice A, 5, does not come from any correct multiplication of the two counts. Choice B, 8, would come from adding instead of multiplying, or from a miscount. Choice C, 12, would result from doubling the correct count, perhaps by mistakenly using 4 shorts options instead of 3.
  9. A student flips a coin and then rolls a standard six-sided number cube (1-6). Which list shows the complete sample space for this compound event?
    Answer: H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6 There are 2 coin outcomes and 6 die outcomes, so the complete sample space has 2 x 6 = 12 ordered pairs: H1 through H6 and T1 through T6, matching choice D. Choice A is missing T1, T3, and T5, leaving only 9 outcomes. Choice B is missing T6, leaving only 11 outcomes. Choice C lists the individual coin and die faces separately instead of the paired outcomes, so it isn't a sample space at all.
  10. A bag has 3 different colored markers: red (R), blue (B), and green (G). A student draws one marker, puts it back, then draws again. Which list is the complete sample space (ordered outcomes)?
    Answer: {RR, RB, RG, BR, BB, BG, GR, GB, GG} Since the marker is put back before the second draw, both draws can be any of the 3 colors, giving 3 x 3 = 9 total possible ordered outcomes. Listing every combination of a first and second draw gives {RR, RB, RG, BR, BB, BG, GR, GB, GG}, matching Choice C. Choice A only lists 6 outcomes and leaves out BR, GR, and GB, so it is missing three possible outcomes. Choice B lists only the outcomes with two different colors and leaves out the doubles like RR, BB, and GG. Choice D mixes single letters with two-letter outcomes instead of showing every draw as a proper pair.
  11. A student draws one colored tile from a bag, does not put it back, and then draws a second tile. The bag contains 2 red tiles (R) and 1 blue tile (B). If outcomes are written in order (first draw, second draw), which list is the complete sample space?
    Answer: {RR, RB, BR} The bag has 2 red tiles and 1 blue tile, and the first tile is not put back before the second draw. If the first draw is red, the second draw can be red or blue, giving outcomes RR and RB. If the first draw is blue, only red tiles remain, giving outcome BR. This means the complete sample space is {RR, RB, BR}, matching Choice B. Choice A is missing the BR outcome, Choice C only shows single draws instead of pairs, and Choice D incorrectly includes BB, which is impossible since there is only one blue tile in the bag.
  12. A student rolls two standard six-sided dice. Use an ordered pair (a,b)(a,b) to mean "die 1 shows aa, die 2 shows bb." Which list correctly gives all outcomes for the event "sum equals 8"?
    Answer: {(2,6),(3,5),(4,4),(5,3),(6,2)}\{(2,6),(3,5),(4,4),(5,3),(6,2)\} This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possibilities, such as for two coins {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches, such as for flipping twice yielding 4 paths. To identify events, read the description like 'both heads' as HH or 'at least one' as HH, HT, TH, then locate and count them. For example, a coin and die list is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for flipping twice shows HH, HT, TH, TT. The correct list is choice A, including all pairs summing to 8 like (2,6) to (6,2). Common errors include missing reverses in B, invalid numbers like 7 in C, or only first die 6 in D. When creating a list, systematically combine all possibilities; for tables, use rows and columns; for trees, branch out with paths as outcomes. To identify events, translate the language like 'sum equals', locate matching outcomes, and count them, avoiding mistakes like incomplete lists, ignoring order, wrong counts, or including invalid outcomes.
  13. A student flips a coin twice. Which set is the complete sample space for this compound event?
    Answer: {HH, HT, TH, TT} Flipping a coin twice produces two independent results, each either heads or tails, giving four total ordered outcomes: HH, HT, TH, and TT, which matches choice C exactly. Choice A only lists the outcomes for a single flip, not two flips. Choice B is missing the TH outcome, incorrectly treating heads then tails and tails then heads as the same result. Choice D only includes the two outcomes where both flips match, leaving out HT and TH entirely. A complete sample space for two flips must include all four ordered pairs, not a partial list.
  14. A student flips a coin twice. How many outcomes are in the event "exactly one head"?
    Answer: 2 This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possibilities, such as for two coins {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches, such as for flipping twice yielding 4 paths. To identify events, read the description like 'both heads' as HH or 'at least one' as HH, HT, TH, then locate and count them. For example, a coin and die list is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for flipping twice shows HH, HT, TH, TT. The correct count is 2 in choice B, for HT and TH. Common errors include 1 for both in A, 3 for at least one in C, or 4 total in D. When creating a list, systematically combine all possibilities; for tables, use rows and columns; for trees, branch out with paths as outcomes. To identify events, translate the language like 'exactly one', locate matching outcomes, and count them, avoiding mistakes like including both or none, wrong event language, tree confusion, or miscounting.
  15. Two number cubes labeled 1-6 are rolled. The ordered pair (a,b)(a,b) means (first cube, second cube). How many outcomes are in the event "the sum is 8"?
    Answer: 5 This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possible outcomes, such as for two coins {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches, such as for flipping a coin twice yielding 4 paths. For example, a coin flip followed by a die roll has a list {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and the event 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for two coin flips shows paths HH, HT, TH, TT. The correct count here is choice B, 5 outcomes where the sum is 8, such as (2,6), (3,5), (4,4), (5,3), (6,2). Common errors include undercounting by missing doubles like (4,4) or overcounting pairs that sum to other values. When creating a table, use rows and columns for each die and mark cells where a + b = 8 to count them. To identify and count, translate 'sum is 8' into pairs and enumerate systematically; mistakes often involve ignoring order or miscounting symmetric pairs.
  16. A student flips a coin and then rolls a standard number cube (1–6). The outcomes can be written like H4 (heads then 4) or T2 (tails then 2). Which set shows the outcomes for the event "heads and an even number"?
    Answer: {H2, H4, H6} This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possible outcomes, such as for two coin flips {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches for each possibility, like four paths for two flips. For example, for a coin flip and die roll, the list is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and the event 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for two flips shows branches leading to HH, HT, TH, TT. The correct representation here is choice B, which accurately identifies the outcomes for heads and an even number: H2, H4, H6. Common errors include listing heads with odds as in A, tails with evens as in C, or all evens regardless of coin as in D. When creating a list, systematically combine all outcomes from each event, then filter for the event. For identifying events, translate the language like 'heads and even' to locate and count matching outcomes, avoiding mistakes like misinterpreting 'and' or including extras.
  17. A student spins a spinner with three equal sections labeled A, B, and C, and then flips a coin. Which list shows the complete sample space for this compound event?
    Answer: {AH,AT,BH,BT,CH,CT} The spinner has 3 equally likely sections, A, B, and C, and the coin has 2 outcomes, heads and tails, so the complete sample space has 3 x 2 = 6 outcomes. Pairing each spinner section with each coin result gives {AH, AT, BH, BT, CH, CT}, matching Choice C. Choice A lists the spinner and coin outcomes separately instead of pairing them together. Choice B is missing the two outcomes involving section C, CH and CT. Choice D leaves out CT, so it only shows 5 of the 6 total outcomes.
  18. A student flips a coin and then rolls a number cube labeled 1-6. Let event E be: The coin shows heads and the number rolled is even. Which set lists all outcomes in E?
    Answer: {H2, H4, H6} The event heads and even only includes outcomes where the coin shows heads and the die shows an even number, which are H2, H4, and H6. This means the correct set is {H2, H4, H6}, matching Choice D. Choice A lists tails paired with even numbers, which does not satisfy the heads part of the event. Choice B includes every outcome with either heads or an even number, rather than only outcomes with both. Choice C lists heads paired with odd numbers, which does not satisfy the even part of the event.
  19. A student rolls two standard number cubes. Outcomes are ordered pairs (first roll, second roll). How many outcomes are in the event "the sum is 8"?
    Answer: 5 This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possible outcomes, such as for two coin flips {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches for each possibility, like four paths for two flips. For example, for a coin flip and die roll, the list is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and the event 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for two flips shows branches leading to HH, HT, TH, TT. The correct representation here is choice B, with 5 outcomes where the sum is 8: (2,6), (3,5), (4,4), (5,3), (6,2). Common errors include undercounting to 4 or 6, or listing total outcomes as 36. When creating a table, fill a 6x6 grid and count cells meeting the condition. For identifying events, translate 'sum is 8' and count carefully, avoiding mistakes like ignoring (4,4) or double-counting.
  20. A student rolls two standard six-sided dice. How many outcomes are in the event "both dice show the same number"?
    Answer: 6 This question tests representing compound sample spaces with lists, tables, or trees, identifying outcomes from everyday language descriptions. Representations include lists that enumerate all possibilities, such as for two coins {HH, HT, TH, TT}, tables in a grid like 6×6 for two dice, or trees with branches, such as for flipping twice yielding 4 paths. To identify events, read the description like 'both heads' as HH or 'at least one' as HH, HT, TH, then locate and count them. For example, a coin and die list is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, and 'heads and even' is {H2, H4, H6} with probability 3/12; a tree for flipping twice shows HH, HT, TH, TT. The correct count is 6 in choice A, for pairs like (1,1) to (6,6). Common errors include counting sums to 12 as 11 in B, doubles plus something as 12 in C, or total as 36 in D. When creating a list, systematically combine all possibilities; for tables, use rows and columns; for trees, branch out with paths as outcomes. To identify events, translate the language like 'both same', locate matching outcomes, and count them, avoiding mistakes like incomplete counts, wrong event language, confusing tree paths, or miscounting table diagonals.

Start this quiz