Understand Probability as Number 0-1 - 7th Grade Math

1. Which value could be a probability? (Remember: probabilities must satisfy $0\le P\le 1$.)
   Answer: $0.75$ This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing a non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 non-Ace, 92% chance); rolling a 7 is impossible P=0 (no 7 on standard die). The correct value is 0.75, as it satisfies 0 ≤ P ≤ 1. Common errors include negatives like -0.2 or >1 like 1.3 or 150 (confusing with 150%), but probabilities can't be outside 0-1. To verify: check if value is in [0,1]; for example, 0.75 is between 0 and 1, interpretable as likely. Mistakes: confusing percent (150% =1.5 >1, invalid) with probability decimals.
2. A probability number line goes from 0 (impossible) to 1 (certain). Where should $0.75$ be located?
   Answer: Three-quarters of the way from 0 to 1, closer to 1 This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0 means impossible, near 0 means unlikely, 1/2 means equally likely as not, near 1 means likely, and 1 means certain, with larger numbers meaning greater likelihood. On the probability scale from 0 to 1, impossible events have P=0 (cannot occur, like rolling a 7 on a standard die), certain events have P=1 (must occur, like rolling a number from 1 to 6 on a die which covers all outcomes), unlikely events have P near 0 (like P=0.1 or 1/10, could happen but probably won't), equally likely events have P=1/2 (50-50, like a coin flip landing heads), and likely events have P near 1 (like P=0.9, probably will occur); larger probabilities indicate greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%), and on a number line, probabilities are plotted from 0 (left, impossible) to 1 (right, certain) with 1/2 at the center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads has P=1/2 (equally likely as tails, 50-50); drawing a non-Ace from a deck has P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 are non-Ace, 92% chance); rolling a 7 is impossible with P=0 (no 7 on a standard die). In this case, 0.75 should be located three-quarters of the way from 0 to 1, closer to 1, indicating likely. A common error is placing it near 0 because 0.75<1/2 (wrong, since 0.75>0.5) or at 1/2 because 0.75=0.5 (incorrect equality), or to the right of 1 because >1 (but probabilities can't exceed 1). To use probability: (1) identify the event and sample space, (2) count favorable and total outcomes, (3) calculate P=favorable/total, (4) interpret (e.g., 0.75 near 1, likely), (5) locate on the 0-1 scale (three-quarters toward 1). Comparing probabilities: larger means more likely (if P(rain)=0.3 and P(sun)=0.7, sun is more likely since 0.7>0.3); complementary events sum to 1 (if P(A)=0.3, P(not A)=0.7); impossible (P=0 exactly, like rolling 7) differs from unlikely (P>0 but near 0, like rolling 1); mistakes include probabilities outside 0-1, wrong likelihood categories, backward comparisons, or confusing percent with probability.
3. A fair coin is flipped once. What is the probability of landing heads?
   Answer: $\frac{1}{2}$ This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0 means impossible, near 0 means unlikely, 1/2 means equally likely as not, near 1 means likely, and 1 means certain, with larger numbers meaning greater likelihood. On the probability scale from 0 to 1, impossible events have P=0 (cannot occur, like rolling a 7 on a standard die), certain events have P=1 (must occur, like rolling a number from 1 to 6 on a die which covers all outcomes), unlikely events have P near 0 (like P=0.1 or 1/10, could happen but probably won't), equally likely events have P=1/2 (50-50, like a coin flip landing heads), and likely events have P near 1 (like P=0.9, probably will occur); larger probabilities indicate greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%), and on a number line, probabilities are plotted from 0 (left, impossible) to 1 (right, certain) with 1/2 at the center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads has P=1/2 (equally likely as tails, 50-50); drawing a non-Ace from a deck has P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 are non-Ace, 92% chance); rolling a 7 is impossible with P=0 (no 7 on a standard die). In this case, the correct probability of landing heads on a fair coin is 1/2, as it is equally likely to happen or not. A common error is claiming P=1 (thinking heads is certain) or P=-1/2 (using negative for unlikely, but probabilities can't be negative), or P=2 (exceeding the maximum of 1). To use probability: (1) identify the event and sample space (coin flip: sample space {heads, tails}), (2) count favorable and total outcomes (heads: 1 favorable, 2 total), (3) calculate P=favorable/total (1/2), (4) interpret (1/2 means equally likely), (5) locate on the 0-1 scale (at center, neither unlikely nor likely). Comparing probabilities: larger means more likely (if P(rain)=0.3 and P(sun)=0.7, sun is more likely since 0.7>0.3); complementary events sum to 1 (if P(A)=0.3, P(not A)=0.7); impossible (P=0 exactly, like rolling 7) differs from unlikely (P>0 but near 0, like rolling 1); mistakes include probabilities outside 0-1, wrong likelihood categories, backward comparisons, or confusing percent with probability.
4. In a carnival game, the probability of winning a small prize is $\frac{5}{12}$, and the probability of winning a large prize is $\frac{1}{6}$. How do these probabilities compare to an event that is neither likely nor unlikely?
   Answer: Both prizes are less likely than an event that is neither likely nor unlikely An event that is neither likely nor unlikely has probability around 1/2 = 6/12. Small prize: 5/12 < 6/12, Large prize: 1/6 = 2/12 < 6/12. Both are less than 1/2. Choice B incorrectly states small prize is more likely. Choices C and D incorrectly compare the probabilities to 1/2.
5. A game show has three doors with prizes. The probability of finding the grand prize behind any door is $\frac{1}{3}$. A contestant argues this means the event is unlikely because "one-third is less than one-half." Is this reasoning correct?
   Answer: Yes, because $\frac{1}{3} \approx 0.33$ is closer to $0$ than to $1$, indicating an unlikely event The contestant's reasoning is correct: since 1/3 is about 0.33, which is closer to 0 than to 1 (well below 0.5), the event is unlikely, matching choice C. Choice A incorrectly assumes that any probability greater than zero makes an event likely, ignoring where the value actually falls on the 0-to-1 scale. Choice B incorrectly claims 0.33 is close enough to 0.5 to count as likely. Choice D incorrectly claims fractional probabilities can't be classified as likely or unlikely; any value between 0 and 1 can be classified this way regardless of its form.
6. A quality control inspector finds that $\frac{11}{50}$ of the products have minor defects. The inspector wants to report this using language that accurately describes the likelihood. Which statement is most appropriate?
   Answer: Products are unlikely to have defects since $\frac{11}{50} = 0.22$ is close to zero When you encounter probability questions, you need to interpret what the numerical value actually means in real-world terms. Converting fractions to decimals often makes this interpretation clearer. First, let's convert $\frac{11}{50}$ to a decimal: $\frac{11}{50} = 0.22 = 22\%$. This means that out of every 100 products, about 22 would have defects while 78 would not. Since significantly more products (78%) are defect-free than defective (22%), products are unlikely to have defects. Answer D correctly identifies this pattern and explains that 0.22 is relatively close to zero. Let's examine why the other choices miss the mark. Choice A makes a logical error—just because a probability is greater than zero doesn't mean an event is likely. Most probabilities are greater than zero, but many events remain unlikely. Choice B incorrectly calls $\frac{11}{50}$ a "large fraction" when it's actually less than half, making it relatively small. Choice C suggests equal chances (50-50), but $\frac{11}{50} = 22\%$ is nowhere near 50%. The key insight is understanding probability ranges: values close to 0 indicate unlikely events, values around 0.5 suggest roughly equal chances, and values close to 1 indicate likely events. Since 0.22 is much closer to 0 than to 0.5, the event is unlikely. Study tip: Always convert fractions to percentages in probability problems—it makes interpretation much clearer. Remember that "unlikely" doesn't mean "impossible," just that the event happens less than half the time.
7. A school survey says $P(\text{students choose pizza})=0.6$ and $P(\text{students choose tacos})=0.4$ (assume these are the only two choices). Which statement is true?
   Answer: Pizza is more likely because $0.6>0.4$. This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0 means impossible, near 0 means unlikely, 1/2 means equally likely as not, near 1 means likely, and 1 means certain, with larger numbers meaning greater likelihood. On the probability scale from 0 to 1, impossible events have P=0 (cannot occur, like rolling a 7 on a standard die), certain events have P=1 (must occur, like rolling a number from 1 to 6 on a die which covers all outcomes), unlikely events have P near 0 (like P=0.1 or 1/10, could happen but probably won't), equally likely events have P=1/2 (50-50, like a coin flip landing heads), and likely events have P near 1 (like P=0.9, probably will occur); larger probabilities indicate greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%), and on a number line, probabilities are plotted from 0 (left, impossible) to 1 (right, certain) with 1/2 at the center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads has P=1/2 (equally likely as tails, 50-50); drawing a non-Ace from a deck has P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 are non-Ace, 92% chance); rolling a 7 is impossible with P=0 (no 7 on a standard die). In this case, the correct statement is that pizza is more likely because 0.6>0.4, as larger probabilities indicate greater likelihood. A common error is saying tacos are more likely because 0.4>0.6 (comparing backwards) or claiming both are equally likely because probabilities must be 0.5 (wrong, they can differ), or stating pizza is certain because 0.6 means 60% which equals probability 60 (confusing percent with scale, and 0.6≠1). To use probability: (1) identify the event and sample space (choices: {pizza, tacos}), (2) count favorable and total outcomes (but here given as probabilities), (3) calculate or use given P, (4) interpret (0.6>0.4, pizza more likely), (5) locate on the 0-1 scale (0.6 closer to 1 than 0.4). Comparing probabilities: larger means more likely (if P(rain)=0.3 and P(sun)=0.7, sun is more likely since 0.7>0.3); complementary events sum to 1 (if P(A)=0.3, P(not A)=0.7); impossible (P=0 exactly, like rolling 7) differs from unlikely (P>0 but near 0, like rolling 1); mistakes include probabilities outside 0-1, wrong likelihood categories, backward comparisons, or confusing percent with probability.
8. Which probability value is not possible (cannot be a probability) because probabilities must satisfy $0\le P\le 1$?
   Answer: $-0.10$ This question tests understanding probability as number 0-1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling 1 on die has P=1/6≈0.17 (1 favorable of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 of 52 non-Ace, 92% chance); rolling 7 impossible P=0 (no 7 on standard die). The value -0.10 is not possible because probabilities must be between 0 and 1 inclusive. Common errors include thinking negative values are allowed for unlikely events (no, use small positive), or overlooking that 0.25, 0.50, and 1 are valid (0.25 unlikely, 0.50 equal, 1 certain). To use probability: (1) check if value is in [0,1], (2) -0.10 <0 so invalid, (3) no calculation, (4) interpret as impossible to have negative P, (5) valid Ps on 0-1 scale only. Mistakes: probabilities outside 0-1 range, like negative or >1.
9. Three events have probabilities of $0.03$, $0.49$, and $0.91$. If these events are arranged from most likely to least likely, what is the correct order?
   Answer: $0.91$, $0.49$, $0.03$ Larger probability numbers indicate greater likelihood. Since 0.91 > 0.49 > 0.03, the correct order from most likely to least likely is 0.91, 0.49, 0.03. Choice A shows least to most likely. Choices C and D have incorrect orderings.
10. Which probability value is NOT possible? (Remember: probabilities must satisfy $0\le P\le 1$.)
    Answer: $-0.2$ This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0 means impossible, near 0 means unlikely, 1/2 means equally likely as not, near 1 means likely, and 1 means certain, with larger numbers meaning greater likelihood. On the probability scale from 0 to 1, impossible events have P=0 (cannot occur, like rolling a 7 on a standard die), certain events have P=1 (must occur, like rolling a number from 1 to 6 on a die which covers all outcomes), unlikely events have P near 0 (like P=0.1 or 1/10, could happen but probably won't), equally likely events have P=1/2 (50-50, like a coin flip landing heads), and likely events have P near 1 (like P=0.9, probably will occur); larger probabilities indicate greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%), and on a number line, probabilities are plotted from 0 (left, impossible) to 1 (right, certain) with 1/2 at the center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads has P=1/2 (equally likely as tails, 50-50); drawing a non-Ace from a deck has P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 are non-Ace, 92% chance); rolling a 7 is impossible with P=0 (no 7 on a standard die). In this case, -0.2 is not possible because probabilities must be between 0 and 1 inclusive, and negatives are invalid. A common error is thinking 0 is not possible (but it is for impossible events) or overlooking that 0.9 and 1/6 are within 0 to 1. To use probability: (1) identify the event and sample space (rolling die: sample space {1,2,3,4,5,6}), (2) count favorable and total outcomes (rolling 1: 1 favorable, 6 total), (3) calculate P=favorable/total (1/6), (4) interpret (≈0.17 near 0, unlikely), (5) locate on the 0-1 scale (in first sixth, closer to 0). Comparing probabilities: larger means more likely (if P(rain)=0.3 and P(sun)=0.7, sun is more likely since 0.7>0.3); complementary events sum to 1 (if P(A)=0.3, P(not A)=0.7); impossible (P=0 exactly, like rolling 7) differs from unlikely (P>0 but near 0, like rolling 1); mistakes include probabilities outside 0-1, wrong likelihood categories, backward comparisons, or confusing percent with probability.
11. A standard deck has 52 cards, including 4 aces. What is the probability of drawing a card that is not an ace, and how should you describe it?
    Answer: $\tfrac{48}{52}\approx 0.92$; likely This question tests understanding probability as number 0-1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling 1 on die has P=1/6≈0.17 (1 favorable of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 of 52 non-Ace, 92% chance); rolling 7 impossible P=0 (no 7 on standard die). The correct probability is 48/52≈0.92, described as likely since it's close to 1. Common errors include P=4/52≈0.08 as very likely (wrong, that's for ace, and it's unlikely), P≈1.50 (invalid >1), or negative P like -0.92 (invalid <0) for unlikely. To use probability: (1) identify event and sample space (drawing card: 52 total), (2) count favorable (non-ace: 48), (3) calculate P=48/52≈0.92, (4) interpret (near 1, likely), (5) locate on 0-1 scale (close to right end). Complementary: P(ace)=4/52≈0.08, P(not ace)=1-0.08=0.92; mistakes include probabilities outside 0-1 or wrong likelihood category.
12. A student claims that an event with probability $0.6$ is unlikely because "it's more than half, so it probably won't happen." What is wrong with this reasoning?
    Answer: Probabilities above $0.5$ indicate likely events, so the event probably will happen When you encounter probability questions, remember that probabilities range from 0 to 1, where values closer to 1 indicate more likely events and values closer to 0 indicate less likely events. The key threshold is 0.5 (or 50%) - this is the dividing line between likely and unlikely events. A probability of 0.6 means the event has a 60% chance of occurring. Since 0.6 is greater than 0.5, this event is actually likely to happen, not unlikely. The student's reasoning contains a fundamental misunderstanding: they correctly identified that 0.6 is "more than half" but then incorrectly concluded this means the event "probably won't happen." In reality, when something is more than half likely, it probably will happen. Looking at the wrong answers: Choice A incorrectly states that 0.6 is impossible - probabilities between 0 and 1 are perfectly valid. Choice B suggests the format matters for determining likelihood, but whether you write 0.6 or 60%, the likelihood remains the same. Choice C contains the same error as the student's original reasoning, claiming unlikely events have probabilities above 0.5, which is backwards. Choice D correctly identifies that probabilities above 0.5 indicate likely events, making this event probable rather than improbable. Study tip: Remember the 0.5 rule - probabilities above 0.5 mean "likely" (more than 50-50 odds), while probabilities below 0.5 mean "unlikely." Don't let the decimal format confuse you; 0.6 = 60% = likely.
13. On the number line shown, which point best represents $P=0.25$?
    Answer: At $0.25$, between 0 and $\dfrac{1}{2}$. This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing a non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 non-Ace, 92% chance); rolling a 7 is impossible P=0 (no 7 on standard die). Correct placement is at 0.25, between 0 and 1/2, as it's closer to 0 (unlikely). Error like at 0.75 thinking 0.25 closer to 1 misreads; or at 1 confusing with certain. To plot: find position on line from 0 to 1; 0.25 is one-quarter from 0, indicating unlikely. Remember: positions reflect likelihood, with left unlikely, right likely.
14. A student draws 1 card from a standard 52-card deck. What is the probability of drawing a non-Ace, and which word best describes this likelihood?
    Answer: $\frac{48}{52}\approx 0.92$, likely This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0 means impossible, near 0 means unlikely, 1/2 means equally likely as not, near 1 means likely, and 1 means certain, with larger numbers meaning greater likelihood. On the probability scale from 0 to 1, impossible events have P=0 (cannot occur, like rolling a 7 on a standard die), certain events have P=1 (must occur, like rolling a number from 1 to 6 on a die which covers all outcomes), unlikely events have P near 0 (like P=0.1 or 1/10, could happen but probably won't), equally likely events have P=1/2 (50-50, like a coin flip landing heads), and likely events have P near 1 (like P=0.9, probably will occur); larger probabilities indicate greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%), and on a number line, probabilities are plotted from 0 (left, impossible) to 1 (right, certain) with 1/2 at the center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads has P=1/2 (equally likely as tails, 50-50); drawing a non-Ace from a deck has P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 are non-Ace, 92% chance); rolling a 7 is impossible with P=0 (no 7 on a standard die). In this case, the correct probability of drawing a non-Ace is 48/52≈0.92, described as likely since it's near 1. A common error is using 4/52≈0.08 and unlikely (confusing with Ace probability) or calling 48/52≈0.92 impossible (wrong interpretation), or stating 48/52=48 and certain (confusing fraction with whole number and mislabeling as certain when it's likely but not 1). To use probability: (1) identify the event and sample space (drawing card: sample space 52 cards), (2) count favorable and total outcomes (non-Ace: 48 favorable, 52 total), (3) calculate P=favorable/total (48/52≈0.92), (4) interpret (near 1, likely), (5) locate on the 0-1 scale (close to 1). Comparing probabilities: larger means more likely (if P(rain)=0.3 and P(sun)=0.7, sun is more likely since 0.7>0.3); complementary events sum to 1 (if P(A)=0.3, P(not A)=0.7); impossible (P=0 exactly, like rolling 7) differs from unlikely (P>0 but near 0, like rolling 1); mistakes include probabilities outside 0-1, wrong likelihood categories, backward comparisons, or confusing percent with probability.
15. Coach Rivera tracks free throw success rates. Player A makes $40\%$ of attempts, Player B makes $\frac{3}{8}$ of attempts, and Player C makes $0.42$ of attempts. Which comparison of their performance is correct?
    Answer: Player C is best with $0.42$, then Player A with $40\%$, then Player B with $\frac{3}{8}$ Converting to decimals: 40% = 0.40, 3/8 = 0.375, and 0.42 = 0.42. Since 0.42 > 0.40 > 0.375, Player C is best. Choice A incorrectly assumes fractions are always largest. Choice C incorrectly treats different values as equal. Choice D compares the numerical digits rather than the actual values.
16. A standard die has 6 equally likely outcomes. What is the probability of rolling a 1? Then classify it as impossible, unlikely, equally likely, likely, or certain.
    Answer: $\dfrac{1}{6}\approx 0.17$ (unlikely) This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing a non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 non-Ace, 92% chance); rolling a 7 is impossible P=0 (no 7 on standard die). The correct probability is 1/6≈0.17 (unlikely) as there is 1 favorable outcome out of 6 equally likely ones. A mistake is claiming P=1.5 (likely), but probabilities can't exceed 1; or confusing with impossible (P=0) when it is possible but unlikely. Steps: (1) sample space {1,2,3,4,5,6}, (2) favorable: {1}, (3) P=1/6, (4) unlikely since <0.5, (5) near 0 on line. Unlikely means possible but low chance, unlike impossible.
17. Maria is designing a spinner for a board game. She wants the probability of landing on the "bonus" section to be unlikely but not impossible. Which probability value would best meet her design goal?
    Answer: $0.15$ A probability near 0 indicates an unlikely event. 0.15 is close to 0, making it unlikely but not impossible. Choice B (0.48) is close to 1/2, indicating neither unlikely nor likely. Choice C (0.52) is also close to 1/2 and slightly likely. Choice D (0.85) is close to 1, indicating a likely event.
18. A standard 52-card deck has 4 Aces. You draw 1 card. What is the probability of drawing a non-Ace? Use the 0 to 1 probability scale and choose the best likelihood description.
    Answer: $\dfrac{48}{52}\approx 0.92$ (likely) This question tests understanding probability as a number between 0 and 1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling a 1 on a die has P=1/6≈0.17 (1 favorable out of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing a non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 out of 52 non-Ace, 92% chance); rolling a 7 is impossible P=0 (no 7 on standard die). Correctly, it's 48/52≈0.92 (likely) since 48 non-Aces out of 52 cards. Error like saying 48 (likely) confuses count with probability; or 4/52 for Ace instead of non-Ace. Process: (1) sample space 52 cards, (2) favorable 48, (3) P=48/52, (4) likely >0.5, (5) near 1. Complementary: P(Ace)=4/52, P(non-Ace)=1 - 4/52=48/52.
19. A weather app says the probability of rain tomorrow is $P(\text{rain})=0.30$. Which statement best interprets this probability on the $0$ to $1$ scale?
    Answer: Rain is somewhat unlikely (less than a 50% chance) This question tests understanding probability as number 0-1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling 1 on die has P=1/6≈0.17 (1 favorable of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 of 52 non-Ace, 92% chance); rolling 7 impossible P=0 (no 7 on standard die). The correct interpretation is that rain is somewhat unlikely with less than a 50% chance, as 0.30 is between 0 and 0.5. Common errors include calling 0.30 certain (wrong, it's not close to 1) or impossible (it's >0), confusing with percent like saying it should be 30 not 0.30, or misinterpreting likelihood (0.30 as likely when it's unlikely). To use probability: (1) identify event (rain tomorrow), (2) the app provides P=0.30, (3) no calculation needed, (4) interpret (0.30 near 0 but >0, unlikely), (5) locate on 0-1 scale (between 0 and 0.5, closer to unlikely). Comparing: larger probability more likely (P(rain)=0.3 vs P(no rain)=0.7, no rain more likely); mistakes include interpreting small P as likely or confusing decimal with whole number.
20. You reach into a box and draw one item. The box contains only pencils and erasers (at least one of each). Which probability best matches the statement: "Drawing a pencil is equally likely as drawing an eraser"?
    Answer: $P(\text{pencil})=\tfrac{1}{2}$ This question tests understanding probability as number 0-1 indicating event likelihood: 0=impossible, near 0=unlikely, 1/2=equally likely as not, near 1=likely, 1=certain, with larger numbers meaning greater likelihood. Probability scale 0 to 1: impossible events P=0 (cannot occur: rolling 7 on standard die), certain events P=1 (must occur: rolling 1-6 on die covers all outcomes), unlikely events P near 0 (like P=0.1 or 1/10: could happen but probably won't), equally likely P=1/2 (50-50: coin flip heads), likely events P near 1 (like P=0.9: probably will occur); larger probability→greater likelihood (P=0.7 means 70% chance, more likely than P=0.3 at 30%); number line: plot probabilities from 0 (left, impossible) to 1 (right, certain), 1/2 at center (neither unlikely nor likely). For example, rolling 1 on die has P=1/6≈0.17 (1 favorable of 6 total outcomes, 17% chance, unlikely—closer to 0 than 0.5); flipping heads P=1/2 (equally likely as tails, 50-50); drawing non-Ace P=48/52≈0.92 (likely—only 4 Aces so 48 of 52 non-Ace, 92% chance); rolling 7 impossible P=0 (no 7 on standard die). The probability P(pencil)=1/2 best matches equally likely as drawing an eraser, implying equal chances. Common errors include P=0 (impossible, but at least one pencil), P=1 (certain, all pencils), or P=100 (invalid, confusing with percent). To use probability: (1) identify event (drawing pencil), sample space (pencils + erasers), (2) for equal likelihood, favorable = half total, (3) P=1/2, (4) interpret (at center, equally likely), (5) locate at 0.5. Complementary: P(eraser)=1 - P(pencil)=0.5; mistakes include wrong values or confusing with certain/impossible.

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