Probability as a Number Between 0 and 1

Lesson 1 of 1

In this lesson:

• Probability is a number from 0 to 1
• Place events on the probability number line
• Explain why probability cannot exceed those bounds

Learning Objectives for This Lesson

1. Define probability as a likelihood measure
2. Write it as a fraction, decimal, or percent
3. Interpret 0, , and 1 on the scale
4. Place events on a probability number line
5. Label events: impossible to certain
6. Explain why probability stays in [0, 1]

What Do You Already Know About Likelihood?

You already use probability language every day:

• "It will probably rain today"
• "There's no way that happens"
• "It's 50-50 — could go either way"

Today's question: Can we replace these vague words with a precise number?

Probability Measures How Likely Events Are

Probability is a number measuring how likely an event is to occur.

• Scale runs from 0 (impossible) to 1 (certain)
• Closer to 0 means less likely; closer to 1 means more likely
• The 0-to-1 scale applies to every event — from coin flips to weather

Probability Scale: Five Landmark Regions

Five landmark regions: impossible, unlikely, equally likely, likely, certain

Five Anchor Examples with Probabilities

Three Ways to Express Probability

Probability can be written as a fraction, decimal, or percent:

All three mean exactly the same thing — choose the form that fits the context.

Check-In: Rolling a Six on a Die

A standard die has faces numbered 1 through 6.

What is the probability of rolling a 6?

• Write it as a fraction and as a decimal
• Assign a label: impossible / unlikely / equally likely / likely / certain

Think before moving on.

Check-In: Probability of Rolling a Six

• Label: Unlikely — below , but not impossible
• Why: Only 1 face out of 6 shows a 6; all faces are equally likely
• Compare: rolling a 7 → P = 0; rolling ≤ 6 → P = 1

From Scale to Line: Ordering Likelihoods

So far we have defined probability and seen five examples.

Next challenge: What if we want to compare many events at once?

The probability number line lets us place, compare, and order events visually — using the same 0-to-1 scale.

Introducing the Probability Number Line

• 0 (left) = Impossible; 1 (right) = Certain
• = equally likely/unlikely — the midpoint divides the line

Placing Events on the Line: Part One

Placing Events on the Line: Part Two

Comparing Probabilities on the Line

Which is more likely — rolling a 3 or picking a vowel?

• Both are in the unlikely zone (below )
• : picking a vowel is slightly more likely

Your Turn: Place Four Events

Draw a 0-to-1 line. Compute and place each point:

1. Roll an even number on a die
2. Random month — it is February
3. Draw a face card (J, Q, K)
4. Random whole number 1–5 is less than 3

Check-In: Which Event Is More Likely?

Answer: Picking a vowel is slightly more likely.

Key lesson: "more likely than" ≠ "likely" — both events are still in the unlikely zone.

Why Must Probability Stay in [0, 1]?

Why can't probability go below 0 or above 1?

• P = 0: Impossible — the event cannot occur. Nothing is "less impossible."
• P = 1: Certain — the event always occurs. Nothing is "more certain."

No probability can fall outside these two anchors.

Fraction Argument: Probability Cannot Exceed One

Probability is often computed as:

• Counts are always non-negative: favorable ≥ 0, so
• Favorable outcomes ≤ total outcomes, so

Identifying Invalid Probability Claims by Range

1. "P(cold this winter) = 1.2" → Invalid: P > 1

2. "P(don't win lottery) = −0.05" → Invalid: P < 0

3. "200% chance the home team wins" → Invalid: Hyperbole; P ≤ 1

Connecting to Complementary Probability Later On

If , what is ?

• An event and its complement always add to 1
• We will formalize this in a later lesson

Check-In: Classify These Probability Claims

Valid (between 0 and 1) or invalid?

Write your answer for each.

Check-In: Valid or Invalid — Answers

1. → Valid ✓
2. → Invalid — exceeds 1; cannot surpass certainty
3. → Invalid — below 0; probability is non-negative

Outside [0, 1] is mathematically wrong.

Applying the Scale: Estimating Real Events

For each event, assign a probability between 0 and 1 and write one sentence defending your choice.

No single "right" answer — the goal is a reasonable, justified estimate.

Estimate the Probability: Five Real Events

Assign a probability from 0 to 1.

1. Rain at least once in the next 30 days
2. Your next meal contains bread or rice
3. A random schoolmate is exactly 13 years old
4. A flipped thumbtack lands point-up
5. A traffic light is green when you arrive

Calculable Probability versus Subjective Estimation

• Calculable: Known equally likely outcomes — die rolls, cards, coins
• Estimated: Depends on location, habits, or data — rain, meal contents
• Data-driven: Thumbtack and traffic light are calculable with experiment data

Probability is always in [0, 1] — but not every probability comes from counting.

Key Takeaways from Today's Lesson

• Probability: a number 0 to 1 measuring likelihood
• 0 = impossible; = equally likely; 1 = certain
• Closer to 1 → more likely; closer to 0 → less likely
• Fraction, decimal, percent — all equivalent
• Outside [0, 1] is mathematically wrong

Watch Out: Five Common Errors

1. ½ = "maybe" — No: equally likely
2. "More likely" ≠ "likely" — 0.3 > 0.1, both unlikely
3. P = 0 = "almost never" — No: impossible
4. Probability = a guess — No: predicts long-run outcomes
5. ¾ = "3, then fail" — a proportion overall

Preview: Experimental Probability in Next Lesson

Next: 7.SP.C.6 — Experimental Probability

• Flip a coin 100 times — how close to 50 heads?
• Run experiments, record outcomes, compute probability from data
• Experimental results approach theoretical values as trials increase