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Lesson 2 of 7

The Mathematics of Maybe: Introduction to Probability · Lesson 2 of 7

The Probability Scale and Certainty

“From impossible to certain, every chance gets an address between 0 and 1.”

Learning Objectives

• Use the probability scale from 0 to 1. • Connect fractions, decimals and percentages to probability. • Distinguish impossible, unlikely, even chance, likely and certain events. • Understand why 99% probability is not a guarantee. • Interpret high-probability statements correctly in daily life.

The Probability Scale

Probability tells us how likely an event is to happen. Its value always lies between 0 and 1, so we can think of probability as a scale that runs from impossible to certain. A probability of 0 means the event cannot happen at all, while a probability of 1 means the event is guaranteed to happen.

Most events are neither completely impossible nor completely certain. Their probabilities lie somewhere between 0 and 1. The closer a probability is to 0, the less likely the event is to occur. The closer it is to 1, the more likely it is to occur. A probability of 0.5 lies exactly in the middle and represents an event that is equally likely to happen or not happen.

Probability scaleLaTeX
ProbabilityMeaning
0Impossible
Between 0 and 0.5Less likely
0.5Even chance
Between 0.5 and 1More likely
1Certain

A probability can be written as a fraction, decimal or percentage. For example, 0.75, 3/4 and 75% all describe the same likelihood.

Probability scale from impossible to certain A horizontal probability scale from zero to one. It shows impossible, less likely, even chance, more likely, and certain, with examples including rolling a three, getting heads, and selecting a red sweet from an all-red bag. Probability Scale From impossible at 0 to certain at 1 Roll a 3 on a die P = 1/6 ≈ 0.17 H Get heads on a coin P = 1/2 = 0.5 Pick a red sweet from an all-red bag P = 1 (certain) 0 0.25 0.5 0.75 1 Impossible Less likely Even chance More likely Certain Probability measures how likely an event is Closer to 0 means less likely; closer to 1 means more likely
Probability scale from impossible to certain
Daily-Life Example: A School Match

Problem
Suppose the estimated probability that your school wins a match is 0.75. What does this mean?

  1. 1.0.75=75%.
  2. 2.Winning is more likely than not winning.
  3. 3.But 0.75 is still less than 1.
  4. 4.So the estimate says a win is likely, not certain.

Why 99% Does Not Mean Guaranteed

A very common misunderstanding is to treat a very high probability as a guarantee. But only probability 1 means certain. A probability of 0.99 means 99 chances out of 100 in a long-run interpretation, while a 1% possibility remains for the event not to happen.

High probability still leaves another possibilityLaTeX

That remaining 0.01 is small, but it is not zero. Therefore the opposite outcome is still possible.

Imagine 100 Identical Cards

Problem
A bag contains 100 identical cards: 99 say WIN and 1 says NOT WIN. You pick one card without looking. Are you guaranteed to pick WIN?

  1. 1.There are 99 favourable cards out of 100, so P(WIN)=99/100=0.99=99%.
  2. 2.But the NOT WIN card is still physically inside the bag.
  3. 3.You could pick that one card on your very first attempt.
  4. 4.Therefore 99% means extremely likely, not guaranteed.
  5. 5.A guarantee would require all 100 cards to say WIN, giving probability 1 or 100%.
Daily-Life Example: Rain Forecast

Problem
A forecast says there is a 99% chance of rain. You go outside and it does not rain. Was the probability statement necessarily wrong?

  1. 1.No. A 99% probability still allows a 1% chance of no rain.
  2. 2.The unlikely outcome can occur on a particular day.
  3. 3.Probability describes uncertainty; it does not force the most likely outcome to happen every time.
  4. 4.The statement would mean rain was overwhelmingly likely, not logically guaranteed.

Rare Does Not Mean Impossible

Students often think that if something has only a 1% chance, it 'basically cannot happen'. Mathematically, that is not correct. Impossible means probability 0. A 1% event is rare, but it can happen. Similarly, a 99% event is highly likely, but it can fail.

The key distinction

Certain means P=1. Impossible means P=0. Any probability strictly between 0 and 1 leaves more than one possible outcome.

One Event vs Many Events

Suppose an event succeeds with probability 99% each time. For one attempt, failure is unlikely. But if the situation is repeated many times, seeing at least one failure becomes much less surprising. Probability is especially useful for understanding repeated events over the long run.

Practice Problems

Practice Problems
  1. Place these probabilities on a scale from impossible to certain: 0, 0.2, 0.5, 0.85, 1.
  2. A bag has 50 slips, 49 marked A and 1 marked B. What is the probability of drawing A? Does that make A certain? Explain.
  3. A weather app gives a 95% chance of rain. Explain in one paragraph why carrying an umbrella is sensible even though rain is not guaranteed.
  4. Give an example of an event with a very small but non-zero probability and explain why it should not be called impossible.
  5. A student says, 'If the probability is 99.9%, the event must happen.' Correct the statement using the probability scale.

Key Takeaways

Key Takeaways

• Probability lies between 0 and 1. • 0 means impossible; 1 means certain. • 0.5 represents an even chance. • A high probability means an event is very likely, not guaranteed. • 99% still leaves a 1% possibility of the opposite outcome. • Rare events are not the same as impossible events.

Coming Next

Next, we learn how to estimate probability objectively using actual experiments and observed data.

Previous · Lesson 1

What Is Probability? and Randomness

Next · Lesson 3

Experimental Probability