Basic Probability Rules
Learning Objectives
- Compute probability with equally likely outcomes
- Describe more than one event
Why This Matters
When Spotify reports that 55% of households subscribe to Netflix and 35% subscribe to Hulu, a content licensing executive can't just add those numbers -- the 18% who subscribe to both would get counted twice, inflating total reach from 72% to 90%. The addition rule corrects for that overlap, and it's the same math behind ad targeting across social platforms, insurance risk pooling, and medical screening coverage. Every time someone asks "how many people does this reach?" about overlapping groups, this formula is the answer.
How to Use This Simulation
- Adjust the P(A), P(B), and P(A∩B) sliders and watch the Venn diagram reshape and the addition rule recalculate in real time.
- Click any region of the Venn diagram to highlight it and see its probability and formula in the results panel.
- Switch between preset scenarios to see how different real-world overlap patterns produce different union probabilities.
- Check the Explanation Panel below -- it updates as you interact and tells you why the numbers change.
What's Happening
Quick Check
A survey finds that 60% of college students use Instagram and 45% use TikTok. A marketing intern adds these and reports that "105% of students use at least one platform." They realize the number can't exceed 100%. What information is the intern missing to calculate the correct percentage?
Try This
Load the "Streaming Services" preset. P(Netflix) = 0.55, P(Hulu) = 0.35, P(both) = 0.18. Use the addition rule to calculate P(Netflix or Hulu) by hand. Then check your answer against the simulation's P(A∪B) display. What percentage of households subscribe to neither service?
A campus survey reports that P(gym membership) = 0.60, P(intramural sports) = 0.45, and P(gym or intramurals) = 0.80. Rearrange the addition rule to solve for P(gym and intramurals). Enter the probabilities into the simulation to verify your answer. In one sentence, explain why knowing P(A∪B) along with the individual probabilities is enough to recover P(A∩B).
A university health center survey finds that 70% of students got a flu shot, 60% got a COVID booster, and 85% got at least one of the two. Determine: (1) the percentage who got both, (2) the percentage who got only a flu shot, and (3) the percentage who got neither. The health center can send one targeted email campaign. Based on the overlap structure, should they target students who got neither vaccination, or students who got only one? Defend your recommendation in two sentences using the probabilities you calculated.
Instructor Notes
Teaching Notes
This simulation works best when you let students drag P(A∩B) before explaining the addition rule. Most students will instinctively try P(A) + P(B) to answer "what's the probability of A or B?" and be surprised when the simulation shows a smaller number. That surprise is the entry point for why subtraction matters -- you're removing the double count.
The forced-minimum-overlap constraint (when P(A) + P(B) > 1) is a powerful teaching moment. Load the Social Media Platforms preset and ask students why P(A∩B) can't go below 0.30. The answer -- that there literally isn't enough room in the sample space for both events without sharing outcomes -- makes the probability axioms concrete.
Common Student Errors
- Adding P(A) + P(B) directly for "or" problems without subtracting the intersection. This is the most common error and the one the simulation is designed to surface.
- Confusing "A and B" notation (P(A∩B)) with the word "and" in plain English. Students sometimes think "A and B" means P(A) + P(B).
- Forgetting that the four regions (A only, B only, A∩B, neither) must sum to 1. This constraint helps students self-check calculations.
- Assuming P(A∩B) can be any value between 0 and 1, without recognizing the upper and lower bounds imposed by P(A) and P(B).
Discussion Questions
- Why does more overlap between two events produce a smaller union probability? Can you think of a real-world situation where this matters?
- If 80% of your class passed the midterm and 75% passed the final, what's the minimum percentage that must have passed both? How do you know?
- A news report says "90% of Americans use the internet and 72% use social media." Can you calculate from just these two numbers what fraction uses both? Why or why not?
Exam Connection
Typical exam questions give P(A), P(B), and one additional value (P(A∩B), P(A∪B), or P(neither)) and ask students to find the missing probabilities. The simulation directly practices all three variants: compute P(A∪B) from the addition rule, work backward to find P(A∩B), and derive P(neither) from P(A∪B). The Stretch challenge specifically practices the backward variant, which students find hardest.