Question

Suppose a fair coin is tossed 3 times. Let E be the event that the first toss is a head and let Fk be the event that there are exactly k heads (0 ≤ k ≤ 3). For what values of k are E and Fk independent?

Answer 1

We first compute the probabilities for various events here as:
P(E) = P(first ross is a head) = 0.5 as first toss is equally likely to be a head or a tail.

P(F₀) = P(no head in 3 tosses) = 0.5³ = 0.125
P(F₁) = P(1 head) = P(HTT, THT, TTH) = 3*0.125 = 0.375
P(F₂ ) = P(2 heads) = P(HHT, HTH, THH) = 0.375,
P(F₃) = P(3 heads) = P(HHH) = 0.125

Now, P(E and F₀) = 0 as there is no outcome when both F₀ and E happens. As this is not equal to P(E)P(F₀) , therefore these 2 events are not independent.

P(E and F₁) = P(HTT) = 0.125 but P(E)P(F₁) = 0.375*0.5 = 0.1875 which is not equal to P(E and F₁), therefore E and F₁ are not independent here.

P(E and F₂) = P(HHT, HTH) = 0.25 but P(E)P(F₂) = 0.375*0.5 = 0.1875 which is not equal to P(E and F₂), therefore E and F₂ are not independent here.

P(E and F₃) = P(HHH) = 0.125 but P(E)P(F₃) = 0.125*0.5 = 0.0625 which is not equal to P(E and F₃), therefore E and F₃ are not independent here.

Therefore there is no value of k for which E and F_k would be independent.