Probability is just counting chances, dressed up in decimals.
Every "likely," "unlikely," and "50/50" you've ever said was probability talk. This lesson turns that everyday instinct into the formal rules — sample space, the addition rule, the multiplication rule, independence, and conditional probability — using dice, marbles, a spinning wheel, and a couple of ASQ-flavoured examples you'll actually meet on the CSSBB and CQE exams.
01 · The idea
What is probability, really?
Probability is a number between 0 (impossible) and 1 (certain) that measures how likely something is to happen. Nothing more mystical than that. "70% chance of rain" means: if today repeated itself many, many times, rain would show up in roughly 7 out of 10 of those repeats.
Slide the marker and see where everyday events land
Two rules make a number a valid probability. First, 0 ≤ P(A) ≤ 1 for any event A — you can't be "120% likely" to do anything. Second, the probabilities of every possible outcome in a situation must add up to exactly 1 — something always happens.
Where do these numbers come from? Three approaches
Every probability you'll ever quote was arrived at one of three ways. The first two are objective — anyone doing the same count or the same experiment gets the same answer. The third is subjective — it's an informed opinion, not a count.
Classical (a priori)
Based on equally likely outcomes you can reason about before running anything — a fair die, a fair coin. This is the favourable/total formula from the last section.
Empirical (relative frequency)
Based on what actually happened in real data — the marble jar and the shift/defect table both used this: observed count ÷ total observations.
Subjective
Based on experience, expert judgment, or partial information — "I'd say there's a 30% chance this supplier ships late." No repeatable experiment backs the number, but it's still a legitimate, usable probability.
02 · The vocabulary
Sample space & events
The sample space is the full list of everything that could happen. Roll two dice and the sample space has 36 equally likely outcomes (6 faces on die 1 × 6 faces on die 2). An event is just a subset of that sample space you happen to care about — "the total is 7," "at least one die shows a 6," and so on.
Two dice, 36 outcomes — click an event to highlight it
Favourable outcomes: 6 / 36 → P = 0.167
Notice the pattern: P(event) = favourable outcomes ÷ total outcomes, as long as every outcome in the sample space is equally likely. That's the classical definition of probability — and it's exactly what the next section builds on.
03 · From counting to observing
The basic formula — and why repetition proves it
That's the theoretical probability — what the math says should happen. The empirical (observed) probability is just: draws that matched A, divided by total draws so far. Draw a jar of marbles enough times and the empirical value creeps toward the theoretical one. That convergence is the Law of Large Numbers, and it's the whole reason sampling and control charts work.
Draw from the jar (4 red, 6 blue — drawn with replacement each time)
Draws: 0
Reds drawn: 0
Empirical P(red): — (theoretical: 0.40)
04 · The easiest shortcut in statistics
The complement rule
"Not A" and "A" always split the sample space in two, so their probabilities must add to 1. This shortcut is a favourite on the CSSBB/CQE exams for "at least one" questions: it's usually far easier to find P(none) and subtract than to add up every "at least one" case directly.
Spin the wheel — then ask "what's the chance it's not red?"
4 sectors: Red 40%, Teal 25%, Amber 20%, Violet 15%.
Landed on: —
Spins so far: 0
Landed "not red": 0
P(red) = 0.40, so P(not red) = 1 − 0.40 = 0.60 — teal, amber, or violet combined.
05 · Combining two events with "or"
The addition rule
You subtract P(A and B) so you don't double-count the overlap. If A and B can never happen together (mutually exclusive), there's no overlap to subtract at all.
A survey of 50 operators — click a region
06 · Combining two events with "and"
The multiplication rule & independence
Two events are independent when knowing one happened tells you nothing new about the other — a coin flip doesn't remember the last flip. They're dependent when it does — drawing a card without putting it back changes the odds for the next draw. P(B | A) is read "the probability of B, given A already happened" — conditional probability.
Follow the tree
Quick reference: OR vs AND
Addition Rule — "OR"
Multiplication Rule — "AND"
07 · A common mix-up
Probability vs. odds
People use "probability" and "odds" interchangeably in conversation, but they're two different numbers built from the same two counts. Probability compares favourable outcomes to all outcomes (part-to-whole). Odds compare favourable outcomes to unfavourable outcomes only (part-to-part).
Probability
Range: 0 to 1 (0% to 100%). Example: rolling a 4 on one die → 1 favourable out of 6 total → P = 1/6 ≈ 16.7%.
Odds
Range: 0 to ∞, usually written as a ratio. Example: rolling a 4 on one die → 1 favourable vs. 5 unfavourable → odds = 1 : 5.
Try it: slide the probability, watch the odds
08 · Putting it to work
On the shop floor: a two-way inspection table
200 finished pipe sections were inspected across two shifts. This is a made-up but realistic example — the exact kind of table you'll build in Excel or Minitab before you ever fit a formal hypothesis test.
| Inspection result | Shift A | Shift B | Total |
|---|---|---|---|
| Defective | 6 | 10 | 16 |
| Good | 94 | 90 | 184 |
| Total | 100 | 100 | 200 |
Reveal the four probabilities examiners love to ask about
Marginal probability — ignore the shift entirely: P(Defective) = 16/200 = 0.08.
Joint probability — both conditions at once: P(Defective and Shift B) = 10/200 = 0.05.
Conditional probability — restrict to Shift B first: P(Defective | Shift B) = 10/100 = 0.10, versus P(Defective | Shift A) = 6/100 = 0.06.
09 · Statistics Implementation
Doing this in Excel and Minitab
Excel functions
| Function | Syntax | Purpose | When to use it |
|---|---|---|---|
| FACT | =FACT(n) | n! — the number of ways to arrange n distinct items in order | Building block for COMBIN/PERMUT, or any "how many arrangements" question |
| PERMUT | =PERMUT(n, k) | Ordered arrangements of k items chosen from n | Order matters — e.g. assigning 3 inspectors to 3 different shifts |
| COMBIN | =COMBIN(n, k) | Unordered selections of k items from n | Order doesn't matter — e.g. how many ways to pick a 5-part sample from a lot of 40 |
| PROB | =PROB(x_range, prob_range, lower, [upper]) | Sums the probabilities of values falling in a range | You already have a discrete probability table and need P(a ≤ X ≤ b) |
| COUNTIF / COUNTIFS | =COUNTIF(range, criteria) | Counts rows meeting one or more conditions | Computing an empirical probability (favourable rows ÷ total rows) from real data |
| RANDBETWEEN | =RANDBETWEEN(bottom, top) | Random integer in a range | Simulating dice rolls, lot sampling, or any discrete uniform draw |
| RAND | =RAND() | Random decimal in [0, 1) | Simulating a fair coin flip (=IF(RAND()<0.5,"H","T")) or any weighted event with a threshold |
Excel use cases
Counting a sampling plan: to find how many distinct 5-part samples can be pulled from a lot of 40, use =COMBIN(40,5) — order doesn't matter, you just need a sample of 5.
Simulating the marble jar from this lesson: put =IF(RANDBETWEEN(1,10)<=4,"Red","Blue") down a column of 500 rows, then =COUNTIF(range,"Red")/500 to watch the empirical probability converge on 0.40, exactly like the widget above.
Building the inspection table: with raw pass/fail-by-shift data in columns, =COUNTIFS(shift_range,"B",result_range,"Defective")/COUNTIF(shift_range,"B") reproduces the conditional probability P(Defective | Shift B) = 0.10 directly from the data instead of a hand-built pivot.
Minitab navigation
Menu path: Calc → Random Data → Bernoulli — generates a column of 0/1 trials at a chosen "success" probability; perfect for simulating a coin flip or a pass/fail inspection outcome, exactly like the empirical side of the marble jar.
Alternative path: Calc → Random Data → Integer — generates uniform random integers over a range (1–6 for a die, for instance), useful for the sample-space and dice examples above.
For counting problems: Calc → Calculator, then search the function list for Combinations and Factorial to reproduce Excel's COMBIN/FACT logic directly on a worksheet.
For a two-way table like the shift/defect example: Stat → Tables → Cross Tabulation and Chi-Square — tabulates counts and lets you display row percentages, which are your conditional probabilities (P(Defective | Shift)).
Which option to use and why: use Random Data commands when you want to simulate outcomes and watch the Law of Large Numbers in action; use the Calculator when you need an exact combinatorial count; use Cross Tabulation whenever real inspection or survey data already exists and you need joint/marginal/conditional probabilities out of it rather than a simulation.
Exam tips
Knowledge check
Check your understanding
Comprehension quiz
Pick the strongest answer, then submit. Explanations appear after you submit.