7.I.1 — Theoretical probability — single event.
Step 01 of 04
Theoretical probability is what SHOULD happen based on the setup of the experiment, assuming all outcomes are equally likely.
P(event)=total number of outcomesnumber of favorable outcomes
Step 02 of 04
Worked examples.
| Roll a 6 on a die | →1/6≈0.167 |
| Roll an even number | →3/6=1/2 |
| Draw a king from a deck | →4/52=1/13 |
| Draw any face card | →12/52=3/13 |
| Pick a vowel from "MATH" | →1/4 (only A is a vowel) |
Step 03 of 04
Probability scale. Always between 0 and 1.
| P=0 | →impossible event |
| P=1 | →certain event |
| P=1/2 | →50/50 — equally likely |
| Never negative | →probabilities can't be less than 0 or more than 1 |
Step 04 of 04
Complement rule. The probability of an event NOT happening is 1−P(event).
P(not 6 on a die)=1−1/6=5/6
Useful when "not" cases are easier to count than "yes" cases.
Key insight
Probability = favorable / total (when outcomes are equally likely). Always between 0 and 1. Complement rule: P(not A)=1−P(A) — great shortcut when "not" is easier to count.
7.I.2 — Experimental probability.
Step 01 of 04
Experimental probability = what ACTUALLY happens when you run the experiment a bunch of times. Compute it from the data.
Pexp=total trialssuccessful trials
Step 02 of 04
Worked example. Flip a coin 50 times, get 23 heads.
Pexp(heads)=23/50=0.46=46%
Theoretical is 50%. The 4-percentage-point gap is normal random variation.
Step 03 of 04
Law of large numbers. As you run MORE trials, experimental probability approaches the theoretical value.
| 10 coin flips | →experimental can range 30–70% |
| 100 flips | →experimental usually within 40–60% |
| 10,000 flips | →experimental almost certainly 49–51% |
| 1,000,000 flips | →experimental virtually identical to 50.000% |
Step 04 of 04
When experimental and theoretical disagree a lot, possibilities are: small sample size (just bad luck), or the assumed model is wrong (the die might be loaded; the coin biased).
Casinos use this — over millions of bets, experimental converges to theoretical, and the house's small theoretical edge becomes their actual edge.
Key insight
Experimental probability is computed from data; theoretical is computed from the setup. They should match in the long run (law of large numbers), but small samples can be surprisingly far off.
7.I.3 — Compound probability — independent events.
Step 01 of 04
Two events are independent if one doesn't affect the other: consecutive coin flips, dice rolls, draws WITH replacement.
P(A and B)=P(A)⋅P(B)
Multiplication rule: probabilities multiply for independent events.
Step 02 of 04
Worked example. Flip a coin AND roll a die. What's the probability of getting heads AND a 6?
P(heads)=1/2,P(6)=1/6
P(both)=1/2⋅1/6=1/12
Step 03 of 04
Two-event tree. Helpful for visualizing.
P(exactly one head) = HT + TH = 1/4+1/4=1/2.
Step 04 of 04
"Or" events use addition, not multiplication. For mutually exclusive (can't happen simultaneously) events:
P(A or B)=P(A)+P(B)
Roll a die — P(1 or 6)=1/6+1/6=1/3.
Key insight
"AND" with independent events → MULTIPLY probabilities. "OR" with mutually exclusive events → ADD. Trees and tables help when the structure gets complicated.
7.I.4 — Random sampling and inferences.
Step 01 of 04
You usually can't measure a whole POPULATION (every voter, every student, every fish in the lake), so you measure a SAMPLE and use it to estimate the population.
For the inference to be valid, the sample must be RANDOM — every member of the population has an equal chance of being chosen.
Step 02 of 04
Random vs. biased samples.
| Survey only your friends | →biased — your friends aren't a random slice |
| Online opt-in poll | →biased — only people who care enough respond |
| Pull names from a hat | →random |
| Computer random number | →random |
| Stratified random (by group) | →random within groups |
Step 03 of 04
Inference example. Survey 50 randomly chosen students, find 32 like pizza. Estimate the percentage of all students who like pizza.
32/50=64%
For a school of 800 students, expect roughly 0.64⋅800=512 to like pizza. The estimate has uncertainty; bigger samples shrink it.
Step 04 of 04
Bigger samples give better estimates. A random sample of 50 is more reliable than 10; 500 is much better still. (Past a few thousand, gains are small relative to the cost.)
But sample QUALITY (truly random) matters more than sample SIZE — a small random sample beats a large biased one.
Key insight
Random sample lets you generalize to the whole population. Biased sample doesn't, no matter how large. Sample percentage estimates population percentage, with uncertainty that shrinks as sample grows.
7.I.5 — Compare two populations using mean and MAD.
Step 01 of 04
To compare two groups: compare their CENTERS (mean or median) and their SPREADS (range, MAD, standard deviation).
MAD = mean absolute deviation, the average distance from the mean. Smaller MAD = data tightly clustered around the mean.
Step 02 of 04
Compute MAD. Step 1: find the mean. Step 2: find each value's distance from the mean (use absolute value). Step 3: average those distances.
Data: 4, 6, 8, 10, 12. Mean = 8.
distances: ∣4−8∣=4,∣6−8∣=2,∣8−8∣=0,∣10−8∣=2,∣12−8∣=4
MAD=(4+2+0+2+4)/5=12/5=2.4
Step 03 of 04
Comparing two classes. Class A test scores: mean 80, MAD 4. Class B: mean 80, MAD 12.
Same mean — both classes performed equally well on average. But Class A's MAD is much smaller — students performed more CONSISTENTLY. Class B has big swings: some students did much better, some much worse.
Step 04 of 04
When centers differ. Class A: mean 75. Class B: mean 85. Class B did better on average. Combined with MAD, you get the full story: "Class B performs about 10 points higher on average, with similar spread" or "Class B is higher on average but more variable."
Key insight
Two numbers describe a dataset: center (mean) and spread (MAD). Same mean with different MAD means same average performance with different consistency. Always compare BOTH when contrasting groups.