Statistics Foundations
Understanding Probability
Probability as a number between 0 (impossible) and 1 (certain).
A Number Between Impossible and Certain
Before reading on, think about these three events and roughly rank how likely each one is: flipping a coin and getting heads, rolling a standard die and getting a 7, and the sun rising tomorrow. Two of these have clear, extreme answers — what number do you think should represent each one?
Definition — Probability
Rolling a 7 on a standard six-sided die is impossible — a probability of 0. The sun rising tomorrow is, for all practical purposes, certain — a probability of 1. Every other event falls somewhere between those two extremes, and the closer to 1, the more likely the event. A fair coin landing heads sits exactly in the middle, at 1/2, since it's genuinely just as likely as not.
For an event where every outcome is equally likely, probability is found the same way you've calculated fractions all along: the number of outcomes that count as a 'success,' divided by the total number of possible outcomes.
Worked Example — Finding the Probability of a Simple Event
Worked Example — Finding the Probability of a Compound Condition
Probability Simulator
Experiment
Run a trial to see the outcome here.
Simulation
Histogram — experimental vs theoretical
Running probability — Law of Large Numbers
Statistics
Mean
— experimental
3.5 theoretical
Variance
— experimental
2.917 theoretical
Standard deviation
— experimental
1.708 theoretical
Trials
0 experimental
— theoretical
Frequency table
| Outcome | Count | Experimental | Theoretical |
|---|---|---|---|
| 1 | 0 | — | 0.167 |
| 2 | 0 | — | 0.167 |
| 3 | 0 | — | 0.167 |
| 4 | 0 | — | 0.167 |
| 5 | 0 | — | 0.167 |
| 6 | 0 | — | 0.167 |
Central Limit Theorem
Draws 300 independent samples of 5trials each from the current experiment, averages each sample, and histograms the resulting sample means — the distribution should look progressively more bell-shaped (normal) as the sample size grows, even though the experiment’s own distribution usually isn’t.
Tip
Common Mistakes
Writing a probability as a number greater than 1, such as reporting a fraction where the numerator exceeds the denominator.
The number of favorable outcomes can never exceed the total number of possible outcomes — recheck the count of favorable outcomes if the resulting fraction is improper.
Miscounting the total number of possible outcomes, especially forgetting to include every category in a mixed group.
Carefully add up every possible outcome in the entire situation before dividing — for the marble example, all 10 marbles count toward the total, not just the ones being asked about.
Key Takeaways
- Probability is a number from 0 (impossible) to 1 (certain) describing how likely an event is.
- For equally likely outcomes, probability equals the number of favorable outcomes divided by the total number of possible outcomes.
- A valid probability always falls between 0 and 1, inclusive.
Summary
Understanding probability as a calculated fraction between 0 and 1 sets up the final lesson's comparison — what actually happens when an experiment is run, versus what theory predicts.
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