Unit 13: Probability and Geometric Modeling · ~15–30 min
Objective
Calculate theoretical and experimental probability for simple events, and apply the complement rule.
What probability measures
Probability is a number from 0 to 1 that describes how likely an event is. A probability of 0 means the event is impossible; a probability of 1 means it's certain. Everything else falls somewhere in between, and can be written as a fraction, decimal, or percent — 1/4, 0.25, and 25% all describe the same likelihood.
Theoretical probability
Theoretical probability is calculated directly from the possible outcomes, assuming every outcome is equally likely:
P(event) = (number of favorable outcomes) ÷ (total number of outcomes)
Rolling a standard number cube, drawing a card, or spinning a fair spinner are all situations where you can count outcomes directly and compute probability without ever running the experiment.
Experimental probability
Experimental (or empirical) probability instead comes from actually performing an experiment and recording results:
P(event) = (number of times the event occurred) ÷ (total number of trials)
The more trials you run, the closer experimental probability tends to drift toward the theoretical value — but with a small number of trials, the two can differ quite a bit just by chance.
The complement rule
The complement of an event A, written "not A," is everything that happens when A doesn't. Since an event either happens or it doesn't, the two probabilities always add to 1:
P(not A) = 1 − P(A)
Every probability is a point on the number line from 0 (impossible) to 1 (certain).
Worked Example 1 · Theoretical probability
ProblemA standard six-sided die is rolled once. Find P(rolling a multiple of 3).
1.The multiples of 3 from 1 to 6 are 3 and 6, so there are 2 favorable outcomes out of 6 total.
2.P = 2/6 = 1/3.
P(multiple of 3) = 1/3
Worked Example 2 · Experimental probability
ProblemA spinner with 4 equal-sized sections (red, blue, green, yellow) is spun 40 times. It lands on red 9 times. Find the experimental probability of red, and compare it to the theoretical probability.
1.Experimental: P(red) = 9/40 = 0.225.
2.Theoretical: since the 4 sections are equal, P(red) = 1/4 = 0.25.
Experimental P(red) = 0.225, close to but not exactly the theoretical 0.25
Worked Example 3 · Complement rule
ProblemThe probability that it rains tomorrow is P(A) = 0.72. Find the probability it does not rain.
1.P(not A) = 1 − P(A) = 1 − 0.72.
P(not A) = 0.28
Guided practice
1.A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. Find the theoretical probability of drawing a blue marble.Show answer
There are 10 marbles total and 3 are blue, so P(blue) = 3/10.
2.A weather app reports a 0.15 probability of snow tomorrow. What is the probability it does not snow?Show answer
By the complement rule, P(not snow) = 1 − 0.15 = 0.85.
3.A basketball player makes 28 out of her last 35 free throws. What is her experimental probability of making a free throw?Show answer
P(make) = 28/35 = 4/5 = 0.8.
4.A standard six-sided die is rolled once. Find P(rolling a number greater than 4).Show answer
The numbers greater than 4 are 5 and 6, so P = 2/6 = 1/3.
5.A student rolls a die 60 times and gets a 6 only 6 times, matching the theoretical probability of 1/6 exactly. Is this always what happens? Explain.Show answer
No — experimental probability is based on actual trials and can vary from the theoretical value by chance, especially with fewer trials. Getting exactly 1/6 is a coincidence here, not a guarantee; over many more trials the experimental result tends to get closer to the theoretical one, but it isn't required to match exactly.