In a fair coin toss, what is the probability that head occurs k times followed by a tail, i.e a tail in k+1 trail, if we toss a coin n times ?
Also, if we modify the question by asking for the probability where an outcome starts with an even number of head, followed by a tail, how do I get the general formula for this?
1 should be obvious from the comments.
It is equal to the probability of any one of these:
T, HHT, HHHHT, HHHHHHT, ...
These are mutually exclusive events, and if the coin in unbiased, the probability is
$\ \ \ \frac 1 2 + \frac1 2 \cdot\frac 1 2 \cdot \frac1 2 + \frac1 2 \cdot\frac 1 2 \cdot \frac1 2 \cdot \frac 1 2 \cdot \frac 1 2 \ \ + \cdot\cdot\cdot $
= $\frac 1 2 (1+ \frac1 {2^2} + \frac 1 {2^4} + \frac1 {2^6} + \cdot\cdot\cdot )$
= $\frac 1 2 \cdot \frac 1 {1-\frac 1 {2^2}} $
= $\frac 2 3$