How to find the conditional probability that five heads are the result when the coin is flipped six times in which the first flip resulted in tails?

Question

My logic: T- Tails,H-Heads

Using the Conditional Probability Formula -

p(E | F)=p(EnF)/p(F)

If you start out with a Tails then that would make:

E={TTTTTT,THTTTT,THHTTT,THHHTT,THHHHT,THHHHH} and

F={HHHHHH,HTHHHH,HTTHHH,HTTTHH,HTTTTH,HTTTTT}

So we need to have 2^5(32) places filled with either heads or tails get

p(EnF)= {THHHHH}= 1/out of the 32 outcomes

p(F)= Set F/ all outcomes =16/32=1/2

Now we plug it back into the formula:

(1/32)/(1/2)=2/32 This answer does seem wrong.

Answer 1

Assuming this is a fair coin, this is how I interpret it: Each toss is independent of the toss(es) that precede it. Therefore, the the first toss (tails) has no influence on the later tosses. As such, the conditional probability = the unconditional probability (see https://en.wikipedia.org/wiki/Conditional_probability). Hence, the probability of the last 5 tosses all being heads is (1/2)^5 = 1/32, irrespective of the outcome of the first toss.