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Put these events in order from least to most likely: rolling a 7 on a standard die, flipping heads on a fair coin, drawing an Ace from a full deck, it raining somewhere in Australia today. Can you place each one on a scale from 0 to 1?
Every probability fits somewhere between 0 (absolutely impossible) and 1 (completely certain). Knowing where events sit on this scale lets us compare likelihoods with precision — not just "pretty likely" but exactly 0.67.
"Unlikely" does NOT mean "impossible". An event with probability 0.01 is very unlikely — but it can still happen. Similarly, "likely" does NOT mean "certain". $P = 0.99$ means the event will almost always happen, but there is still a 1% chance it won't. Only $P = 0$ is impossible and only $P = 1$ is certain. Every value in between represents genuine uncertainty.
| Label | P value | Example |
|---|---|---|
| Impossible | 0 | Rolling a 7 on a standard die |
| Unlikely | 0 < P < 0.5 | Drawing an Ace from a deck (≈ 0.077) |
| Even Chance | 0.5 | Flipping heads on a fair coin |
| Likely | 0.5 < P < 1 | Rolling a number greater than 2 on a die (≈ 0.67) |
| Certain | 1 | It being daytime somewhere on Earth right now |
Probability values can be written three ways — always be comfortable switching between them:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 4/52 = 1/13 | 0.077 | 7.7% |
To convert: fraction → decimal: divide numerator by denominator. Decimal → percentage: multiply by 100.
Calculate the probability first, then decide where it sits:
When $P(A) > P(B)$, event A is more likely than event B. The probability scale is just a number line — bigger numbers sit further to the right.
Example — ordering by likelihood:
$P(\text{roll} = 1) = \dfrac{1}{6} \approx 0.17 \qquad P(\text{roll} > 2) = \dfrac{4}{6} \approx 0.67$
So rolling $> 2$ is much more likely than rolling exactly 1. In order from least to most likely:
Roll 7 (impossible, $P=0$) < Draw Ace ($P \approx 0.08$) < Roll = 1 ($P \approx 0.17$) < Flip heads ($P = 0.5$) < Roll $> 2$ ($P \approx 0.67$) < Rain in AU today ($P = 1$)
The Probability Scale:
$$0 \longleftarrow \text{impossible} \quad \text{unlikely} \quad \underbrace{0.5}_{\text{even chance}} \quad \text{likely} \quad \text{certain} \longrightarrow 1$$
Converting: fraction $\to$ decimal (÷ bottom by top), decimal $\to$ % (× 100).
$P(A) > P(B)$ means A is more likely than B. Order events by their numerical values.
An event has probability 0. This means the event is:
A probability of 0.5 on the scale means the event is:
Which of these events is the least likely?
An event has probability 0.8. Which label best describes it on the probability scale?
Which of the following events is certain?
Q6. Five events have the following probabilities: 0, 0.1, 0.5, 0.8, 1.
(a) Draw a number line from 0 to 1 and place all five values on it.
(b) Label each with the correct scale descriptor (impossible / unlikely / even chance / likely / certain).
(c) Which two values are complements of each other? Explain why.
Q7. Convert the following probabilities between all three forms (fraction, decimal, percentage):
(a) $\dfrac{3}{4}$ (b) 0.6 (c) 35% (d) $\dfrac{7}{20}$ (e) 0.08
For each one, also state whether the event is impossible, unlikely, even chance, likely, or certain.
Q8. A bag contains some coloured marbles. Consider these four events:
Event A: drawing a red marble. Event B: drawing a marble that is NOT blue. Event C: drawing a marble that is purple (when there are no purple marbles). Event D: drawing any marble at all.
(a) Write the probability of Event C and classify it on the probability scale.
(b) Write the probability of Event D and classify it.
(c) If P(red) = 3/10, classify Event A and find P(not red).
(d) If P(blue) = 2/10, find P(not blue) and classify Event B.
(a) Number line: 0 ————— 0.1 ————— 0.5 ————— 0.8 ————— 1
(b) 0 = impossible; 0.1 = unlikely; 0.5 = even chance; 0.8 = likely; 1 = certain.
(c) 0.1 and 0.9 are complements (0.1 + 0.9 = 1), but 0.9 is not listed. From the given list, 0 and 1 are complements — if an event is impossible, its complement (the event NOT happening) is certain.
(a) 3/4 = 0.75 = 75% — likely.
(b) 0.6 = 3/5 = 60% — likely.
(c) 35% = 0.35 = 7/20 — unlikely.
(d) 7/20 = 0.35 = 35% — unlikely.
(e) 0.08 = 2/25 = 8% — unlikely.
(a) P(purple) = 0 — impossible (no purple marbles in the bag).
(b) P(any marble) = 1 — certain (every draw will produce some marble).
(c) P(red) = 3/10 = 0.3 — unlikely. P(not red) = 1 − 3/10 = 7/10 = 0.7.
(d) P(blue) = 2/10 = 0.2. P(not blue) = 1 − 0.2 = 0.8 — likely.
A spinner is divided into 8 equal sections, each labelled with a number. You want to design it so that:
• $P(\text{even number}) = 0.5$
• $P(\text{number} = 1) = 0.125$
• $P(\text{number} = 8) = 0.25$
(a) How many sections show the number 1? Justify using the probability.
(b) How many sections show the number 8?
(c) P(even) = 0.5 means 4 sections show even numbers. Two sections show 8 (which is even). What other even numbers could fill the remaining 2 even sections?
(d) Is there more than one valid design for this spinner? Explain.
(e) What is $P(\text{not } 8)$ on your spinner?