The Probability Scale
Place any event on a scale from 0 to 1, and express it as a fraction, decimal, or percentage.
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Why does this matter?
Doctors say a surgery has an 80% success rate. Weather forecasters say there's a 0.3 chance of rain. Bookmakers express odds as fractions like 3/4. All of these are the same probability just written in different forms. Understanding the probability scale means you can interpret any of these at a glance and compare them confidently.
The Big Idea
All probabilities sit on a scale from 0 to 1. The words we use, impossible, unlikely, even chance, likely, certain, match specific positions on this scale. Any probability can be written as a fraction, decimal, or percentage. These are just three ways to express the same number.
$$\frac{3}{4} = 0.75 = 75\%$$
Key Terms
Spot the Trap
- Thinking probabilities are "out of 100": P = 0.7 does NOT mean 0.7 out of 100, it means 70 out of 100 (i.e., 70%). Don't write P = 70 just because the chance is 70%.
- Placing "likely" and "certain" at the same position: "Likely" means more probable than not (e.g., P = 0.8), while "certain" means P = 1 exactly. They are different positions on the scale.
- Confusing 50-50 with 50: An even chance is P = 0.5, not P = 50. Probability is never written as "50", that would be 5000%!
Explore the Concept
Reading the Probability Scale
+5 XPThe probability scale runs from 0 (impossible) to 1 (certain). Every event sits at exactly one point on this scale. Words like "likely" and "unlikely" describe regions, not exact values.
0: Impossible, rolling a 9 on a standard die.
0 to 0.5 (exclusive): Unlikely, drawing a spade from a shuffled deck (P = 1/4 = 0.25).
0.5: Even chance, flipping heads on a fair coin.
0.5 to 1 (exclusive): Likely, rolling a number less than 5 on a die (P = 4/6 ≈ 0.67).
1: Certain, rolling a number less than 10 on a standard die.
Converting Between Forms
+5 XPProbability can be expressed in three equivalent forms. You need to move fluently between them.
Fraction → Decimal: Divide top by bottom. $\frac{3}{4} = 3 \div 4 = 0.75$
Decimal → Percentage: Multiply by 100. $0.75 \times 100 = 75\%$
Percentage → Decimal: Divide by 100. $75\% \div 100 = 0.75$
Decimal → Fraction: Write over 10, 100, etc. $0.6 = \frac{6}{10} = \frac{3}{5}$
Always simplify fractions where possible. Check: does your final answer sit sensibly between 0 and 1?
Placing Events on the Scale
+5 XPTo place an event on the probability scale: calculate its probability as a decimal, then locate that value between 0 and 1. Label the scale position clearly.
Rolling an odd number on a die: {1,3,5} → P = 3/6 = 0.5 → even chance (midpoint).
Drawing a heart from a deck: 13/52 = 1/4 = 0.25 → unlikely (1/4 of the way from 0).
Picking a consonant from {B,C,D}: 3/3 = 1 → certain (right end).
When comparing events, convert to the same form (all decimals is often easiest) before comparing positions.
Worked Examples
Convert P = 3/4 to Decimal and Percentage
+15 XPDivide the numerator by the denominator.
$$\frac{3}{4} = 3 \div 4 = 0.75$$
Multiply the decimal by 100.
$$0.75 \times 100 = 75\%$$
P = 0.75 sits between 0.5 and 1, it is in the likely region of the probability scale, three-quarters of the way from impossible to certain.
Place 5 Events on the Probability Scale
+15 XPA: Rolling a 1 on a die → P = 1/6 ≈ 0.17
B: Flipping tails on a coin → P = 1/2 = 0.5
C: Drawing a red card from a deck → P = 26/52 = 0.5
D: Rolling less than 6 on a die → P = 5/6 ≈ 0.83
E: Rolling 7 on a standard die → P = 0
E (0) < A (≈0.17) < B = C (0.5) < D (≈0.83)
Note: B and C are at the same position, both have P = 0.5.
E → Impossible | A → Unlikely | B, C → Even chance | D → Likely
None of these events are certain (P = 1), so no event sits at the right endpoint.
Convert 35% to Decimal and Fraction
+15 XPDivide by 100 to convert percentage to decimal.
$$35\% \div 100 = 0.35$$
Write as a fraction over 100, then simplify.
$$0.35 = \frac{35}{100} = \frac{7}{20}$$
GCF of 35 and 100 is 5: 35 ÷ 5 = 7, 100 ÷ 5 = 20.
P = 0.35 < 0.5, so this event is in the unlikely region, it could happen, but probably won't.
Common Pitfalls
- P = 50% written as P = 50: Probability is always 0 to 1. Write P = 0.5 or P = 50%, never P = 50 on its own.
- Inconsistent form in answers: If a question says "express as a decimal," don't answer with a percentage. Match the form requested.
- Assuming 50-50 means unlikely + likely = 1: Two complementary events share the scale symmetrically, but unequal events can each be "likely", their probabilities just must sum to 1.
Copy-ready Notes
Probability scale: 0 (impossible) → 0.5 (even chance) → 1 (certain).
Scale words: impossible (P=0), unlikely (0<P<0.5), even chance (P=0.5), likely (0.5<P<1), certain (P=1).
Converting: Fraction → Decimal (divide). Decimal → % (× 100). % → Decimal (÷ 100). Decimal → Fraction (write over 100, simplify).
Key rule: Probability is ALWAYS between 0 and 1 inclusive. Never write probability as a raw percentage number without the % symbol.
Brain Trainer
Convert 2/5 to a decimal and a percentage.
Decimal: 2 ÷ 5 = 0.4
Percentage: 0.4 × 100 = 40%
Convert P = 0.35 to a percentage and a simplified fraction.
Percentage: 0.35 × 100 = 35%
Fraction: 35/100 = 7/20 (divide by GCF 5)
Give one example of an event for each of these scale words: impossible, unlikely, even chance, likely, certain.
Impossible (P=0): Rolling a 10 on a standard die.
Unlikely (0 < P < 0.5): Rolling a 1 on a standard die (P = 1/6 ≈ 0.17).
Even chance (P=0.5): Flipping heads on a fair coin.
Likely (0.5 < P < 1): Rolling a number less than 5 on a die (P = 4/6 ≈ 0.67).
Certain (P=1): Rolling a whole number less than 10 on a standard die.
Is P(rolling ≤ 4 on a standard die) closer to 0 or 1? Explain and calculate.
Favourable outcomes: {1, 2, 3, 4} → 4 outcomes. Total: 6.
P = 4/6 = 2/3 ≈ 0.67
Since 0.67 > 0.5, it is closer to 1. This event is in the "likely" region of the scale.
Multiple Choice
1. On the probability scale, an event with an "even chance" is located at:
- 0
- 0.25
- 0.5
- 1
2. P(event) = 3/8. Which scale word best describes this event?
- Unlikely
- Even chance
- Likely
- Certain
3. A weather forecast gives a 65% chance of rain. What is this as a decimal?
- 65
- 6.5
- 0.065
- 0.65
4. Which fraction is equivalent to the decimal 0.4?
- 4/100
- 2/5
- 4/5
- 1/4
5. Which event is closest to "certain" on the probability scale?
- P = 3/5
- P = 70%
- P = 0.9
- P = 4/6
Short Answer
Q6. A spinner is divided into 10 equal sections: 3 red, 4 blue, 3 green.
(a) Calculate P(blue) as a fraction, decimal, and percentage.
(b) Where does P(blue) sit on the probability scale? Name the scale word.
(c) Calculate P(not blue).
Q7. Order the following probabilities from smallest to largest and label each with a scale word:
A = 7/10 B = 0.15 C = 50% D = 1/3 E = 0
Q8. A student writes: "The probability of passing the exam is 85%, so P(failing) = 15%."
(a) Rewrite both probabilities as decimals.
(b) Is the student's reasoning correct? Explain using the complement rule.
Check Your SAQ Answers
Q6.
(a) P(blue) = 4/10 = 2/5 = 0.4 = 40%
(b) P = 0.4 < 0.5 → this is in the unlikely region of the scale.
(c) P(not blue) = 1 − 0.4 = 0.6
Q7. Convert all to decimals: A = 0.7, B = 0.15, C = 0.5, D ≈ 0.333, E = 0.
Order: E (0) < B (0.15) < D (0.333) < C (0.5) < A (0.7)
Scale words: E = impossible, B = unlikely, D = unlikely, C = even chance, A = likely
Q8.
(a) P(pass) = 85% ÷ 100 = 0.85; P(fail) = 15% ÷ 100 = 0.15
(b) Yes, the student is correct. Since "passing" and "failing" are complementary events (you either pass or fail, no other option), P(fail) = 1 − P(pass) = 1 − 0.85 = 0.15. The two probabilities sum to 1.
Stretch, The Probability Reporter
+25 XPA sports journalist writes these three probability statements about tomorrow's match:
- "Team A has a 3/5 chance of winning."
- "There is a 35% chance of a draw."
- "Team B winning is unlikely, with odds of 1 in 10."
(a) Convert all three probabilities to decimals.
(b) Do all three probabilities sum to 1? If not, what is wrong?
(c) The journalist later says: "Actually, Team A winning or Team B winning are the only two possible outcomes, no draw is possible." Recalculate the probability of Team B winning using the complement rule.
(d) With only two possible outcomes (A wins or B wins), what must P(A wins) + P(B wins) equal? Use your answer from (c) to verify.
Lesson Summary
- The probability scale runs from 0 (impossible) to 1 (certain).
- Scale words: impossible, unlikely, even chance (0.5), likely, certain.
- Any probability can be expressed as a fraction, decimal, or percentage.
- Convert: fraction → divide; decimal → multiply by 100 for %; % → divide by 100 for decimal.
- Probability is never a whole number like "50", it's always between 0 and 1.
True or False Revisit
A probability of 0.5 means an even chance, equally likely to happen or not.
To convert a decimal to a percentage you divide by 100.
P = 0.7 is in the "likely" region of the probability scale.
Writing P = 50 is acceptable when the probability is 50%.