7th Grade Math Quiz: Represent Sample Spaces For Compound Events

What this quiz covers

This quiz focuses on Represent Sample Spaces For Compound Events, giving you a quick way to practice the rules, question types, and explanations that matter most for 7th Grade Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 3 favorable outcomes out of 10 total possible drawing combinations
2. 6 favorable outcomes out of 12 total possible drawing combinations
3. 4 favorable outcomes out of 14 total possible drawing combinations
4. 4 favorable outcomes out of 12 total possible drawing combinations (correct answer)

Explanation: 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}:

Question 2

1. The sample space contains 16 outcomes, and exactly 4 outcomes satisfy the given conditions (correct answer)
2. The sample space contains 12 outcomes, and exactly 3 outcomes satisfy the given conditions
3. The sample space contains 16 outcomes, and exactly 6 outcomes satisfy the given conditions
4. The sample space contains 8 outcomes, and exactly 2 outcomes satisfy the given conditions

Explanation: Sample space = 2 × 2 × 4 = 16 outcomes (each coin has 2 outcomes, wheel has 4). For exactly one head: either HT or TH with any wheel number. Numbers less than 3 are 1 and 2. Favorable outcomes: (H,T,1), (H,T,2), (T,H,1), (T,H,2) = 4 outcomes. Choice B undercounts the sample space. Choice C overcounts favorable outcomes. Choice D undercounts both.

Question 3

1. The sample space has 30 outcomes, with exactly 4 outcomes satisfying both conditions simultaneously (correct answer)
2. The sample space has 25 outcomes, with exactly 4 outcomes satisfying both conditions simultaneously
3. The sample space has 30 outcomes, with exactly 6 outcomes satisfying both conditions simultaneously
4. The sample space has 11 outcomes, with exactly 2 outcomes satisfying both conditions simultaneously

Explanation: Sample space = 6 die outcomes × 5 card outcomes = 30 total outcomes. Die multiples of 3 are 3 and 6. Vowel cards are A and E. Favorable outcomes: (3,A), (3,E), (6,A), (6,E) = exactly 4 outcomes. Choice B undercounts the sample space. Choice C overcounts favorable outcomes. Choice D severely undercounts the sample space.

Question 4

1. {H, T}
2. {HH, HT, TT}
3. {HH, HT, TH, TT} (correct answer)
4. {HH, TT}

Explanation: 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.

Question 5

1. 12
2. 10
3. 6
4. 8 (correct answer)

Explanation: The spinner has 4 equally likely sections and the coin has 2 possible results, so the sample space has 4 times 2 equals 8 total outcomes, matching choice D. Choice B (10) and choice A (12) overcount, likely from adding instead of multiplying the two events. Choice C (6) undercounts, possibly from forgetting to pair every spinner section with both coin results. Listing all outcomes systematically, such as A-Heads, A-Tails, B-Heads, B-Tails, and so on through D-Tails, confirms there are exactly 8 pairs.

Question 6

1. {HH, HT, TH} (correct answer)
2. {HT, TH}
3. {TT}
4. {HH}

Explanation: The sample space for flipping a coin twice is {HH, HT, TH, TT}. The event "at least one head" includes every outcome that has one or more heads, which means every outcome except TT. That gives {HH, HT, TH}, matching Choice A. Choice B is missing HH, so it leaves out an outcome that does have a head. Choice C, {TT}, is the only outcome with no heads at all, which is the opposite of what the event asks for. Choice D, {HH}, only includes the outcome with two heads and misses the outcomes with exactly one head.

Question 7

1. 9 (correct answer)
2. 12
3. 6
4. 3

Explanation: 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.

Question 8

1. 6
2. 4
3. 36
4. 5 (correct answer)

Explanation: 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.

Question 9

1. 12
2. 6
3. 9 (correct answer)
4. 3

Explanation: 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.

Question 10

1. {HT, TH, TT}
2. {HH, HT, TT}
3. {HH, HT, TH, TT} (correct answer)
4. {H, T}

Explanation: Flipping a coin twice creates 2 x 2 = 4 possible outcomes, since each flip can land heads or tails independently. Listing every combination gives {HH, HT, TH, TT}, which matches Choice C. Choice A is missing HH, and Choice B is missing TH, so both leave out a possible outcome. Choice D only lists the outcomes for a single flip, not two.

Question 11

1. {A,B,C,H,T}
2. {AH,AT,BH,BT}
3. {AH,AT,BH,BT,CH,CT} (correct answer)
4. {AH,BH,CH,AT,BT}

Explanation: 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.

Question 12

1. {H2, H4, H6, T2, T4, T6}
2. {H2, H4, H6} (correct answer)
3. {H1, H3, H5}
4. {T2, T4, T6}

Explanation: 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.

Question 13

1. H1,H2,H3,H4,H5,H6,T2,T4,T6
2. H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5
3. 1,2,3,4,5,6,H,T
4. H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6 (correct answer)

Explanation: 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.

Question 14

1. {HT, TH}
2. {HH, HT, TH} (correct answer)
3. {HH}
4. {HH, HT, TH, TT}

Explanation: 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 representation here is choice C, which includes all outcomes with at least one head: HH, HT, TH. Common errors include only both heads in A, only one head in B, or the full sample space in D. When creating a list, systematically combine outcomes and filter for the event, like excluding TT for 'at least one head.' To identify events, translate phrases like 'at least one' to include all qualifying outcomes and count them; mistakes often involve confusing 'at least' with 'exactly' or omitting ordered variations.

Question 15

1. {BB}
2. {RB, BR} (correct answer)
3. {RR, RB, BR}
4. {RR}

Explanation: 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 A, identifying RB and BR as exactly one blue. Common errors include impossible BB as in B, all outcomes as in C, or none as in D. When creating a list, systematically list possible sequences considering without replacement. For identifying events, translate 'exactly one' to match outcomes with one B, avoiding including zero or two.

Question 16

1. (2, 6) (correct answer)
2. (5, 4)
3. (6, 1)
4. (3, 6)

Explanation: 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 A, as (2,6) sums to 8 in the table. Common errors include pairs summing to 9 like B or C, or 7 like D. When creating a table, use rows and columns for each die, with cells as outcomes. For identifying events, translate 'sum is 8' to locate matching pairs, avoiding miscounting sums or confusing table positions.

Question 17

1. {(5,6), (6,5), (6,6)} (correct answer)
2. {(4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}
3. {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
4. {(5,6), (6,6)}

Explanation: 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 representation here is choice A, listing all three outcomes where sum >10: (5,6), (6,5), (6,6). Common errors include including sums of 10 or less in B, missing (6,5) in C, or listing sum=7 in D. When creating a list, systematically check pairs where a + b >10, considering order. To identify events, translate 'greater than 10' to qualifying pairs and ensure completeness; mistakes often involve wrong inequalities or omitting doubles like (6,6).

Question 18

1. $\{H2,H4,H6\}$ (correct answer)
2. $\{H1,H2,H3,H4,H5,H6\}$
3. $\{H2,H4,H6,T2,T4,T6\}$
4. $\{H1,H3,H5\}$

Explanation: 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.

Question 19

1. 1
2. 2 (correct answer)
3. 3
4. 4

Explanation: 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.

Question 20

1. 12
2. 11
3. 6 (correct answer)
4. 36

Explanation: 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.