Probability Equation Solver

Problem 2.1 • Find P(A) and P(B) given P(A∩B) and P(A∪B)

📝 Input Known Values

System of Equations

P(A) + P(B) − P(A∩B) = P(A∪B)
⟹ P(A) + P(B) = 0.90

💡 Key Insight

With only P(A∩B) and P(A∪B), we can find P(A)+P(B) but not individual values. We need one more constraint (like independence) to solve uniquely.

📊 Solution

Infinite Solutions

Any (P(A), P(B)) pair that sums to the required value works.

P(A) + P(B)
0.90

📐 Derivation

1. Start with: P(A∪B) = P(A) + P(B) − P(A∩B)
2. Rearrange: P(A) + P(B) = P(A∪B) + P(A∩B)
3. Substitute: P(A) + P(B) = 0.70 + 0.20 = 0.90

🎯 If Independent (A ⊥ B)

With independence: P(A∩B) = P(A)·P(B), we get a quadratic!

Let s = P(A) + P(B), p = P(A)·P(B)
Then P(A), P(B) are roots of:
x² − sx + p = 0
P(A)
0.50
P(B)
0.40