Coin Toss Independence

Problem 3.2 • When are "First coin is Heads" and "Exactly k Heads" independent?

🎲 Outcome Space (Ω)

Showing all outcomes for n=3.

⚙️ Parameters

Number of Tosses (n) 3
Target Heads (k) 2
Event A: First toss is Head (H...)
Event B: Exactly k heads total

📊 Probabilities

P(A) 0.500
P(B) 0.375
P(A ∩ B) 0.125
P(A) × P(B) 0.188

Dependent

P(A∩B) ≠ P(A)P(B)

💡 Insight

For "First Toss H" and "k Heads" to be independent, we essentially need knowing the first coin is H to not change the probability of getting k heads total. This happens exactly when n = 2k.