02 // Visual Math & Finance SolversP(A | B) = P(A ∩ B) / P(B), P(B | A) = P(A ∩ B) / P(A)
Visual Conditional Probability $P(A|B)$ & Independence Venn Lab
Conditional probability $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ restricts the active sample space to region $B$ and measures the fraction of $B$'s area occupied by the overlap lens $A \cap B$.
P(A | B) = P(A ∩ B) / P(B), P(B | A) = P(A ∩ B) / P(A)Inputs & Parameters (4)
5±1200
5±1200
0±1150
50±5500
Union |A ∪ B|
90
45% of Universe |U| = 200
Intersection |A ∩ B|
20
Jaccard Index J = 22.2%
Conditional P(A | B)
40%
20 / 50 | P(B|A) = 33.3%
Symmetric Diff |A Δ B|
70
Only A: 40, Only B: 30, Exterior: 110
20-Row Comparison Table
Compare across:
| # | Event A Count n(A) | Union |A ∪ B| | P(A | B) | Symm Diff |A Δ B| |
|---|---|---|---|---|
| 01 | 5 | 50 | 10% | 45 |
| 02 | 15 | 50 | 30% | 35 |
| 03 | 26 | 56 | 40% | 36 |
| 04 | 36 | 66 | 40% | 46 |
| 05 | 46 | 76 | 40% | 56 |
| 06 | 56 | 86 | 40% | 66 |
| 07 | 67 | 97 | 40% | 77 |
| 08 | 77 | 107 | 40% | 87 |
| 09 | 87 | 117 | 40% | 97 |
| 10 | 97 | 127 | 40% | 107 |
| 11 | 108 | 138 | 40% | 118 |
| 12 | 118 | 148 | 40% | 128 |
| 13 | 128 | 158 | 40% | 138 |
| 14 | 138 | 168 | 40% | 148 |
| 15 | 149 | 179 | 40% | 159 |
| 16 | 159 | 189 | 40% | 169 |
| 17 | 169 | 199 | 40% | 179 |
| 18 | 179 | 209 | 40% | 189 |
| 19 | 190 | 220 | 40% | 200 |
| 20 | 200 | 230 | 40% | 210 |
Was this tool helpful?
Key Analytical Takeaways
- 01In general, $P(A \mid B) \ne P(B \mid A)$ because they divide the same overlap $|A \cap B|$ by different denominators ($|B|$ vs. $|A|$).
- 02Events $A$ and $B$ are statistically independent when $P(A \cap B) = P(A) \cdot P(B)$, which means knowing $B$ occurred does not change $P(A \mid B) = P(A)$.
- 03When $|A \cap B| = 0$, the circles are disjoint (mutually exclusive) and $P(A \mid B) = 0$.
Frequently Asked Questions
Are mutually exclusive events independent?+
No—if two non-empty events are mutually exclusive (disjoint, A ∩ B = ∅), knowing that B occurred guarantees A did NOT occur (P(A|B) = 0), making them maximally dependent.
How does this Venn diagram connect to Bayes' Theorem?+
Since |A ∩ B| = |B|·P(A|B) = |A|·P(B|A), dividing by |B| gives P(A|B) = P(B|A)·P(A) / P(B).
What is the difference between independent events and mutually exclusive events?+
Mutually exclusive events cannot occur together (P(A ∩ B) = 0, non-overlapping circles), whereas independent events satisfy P(A ∩ B) = P(A)·P(B), meaning knowing B occurred leaves P(A) unchanged.