The problem statement, all variables and given/known data:
Suppose a fair coin is tossed $n$ times. Find simple formulae in terms of $n$ and $k$ for
a) $P(k-1 \mbox{ heads} \mid k-1 \mbox{ or } k \mbox{ heads})$
b) $P(k \mbox{ heads} \mid k-1 \mbox{ or } k \mbox{ heads})$
Relevant equations:
$P(k \mbox{ heads in } n \mbox{ fair tosses})=\binom{n}{k}2^{-n}\quad (0\leq k\leq n)$
The attempt at a solution:
I'm stuck on the conditional probability. I've dabbled with it a little bit but I'm confused what $k-1$ intersect $k$ is. This is for review and not homework.
The answer to a) is $k/(n+1)$.
I tried $P(k-1 \mbox{ heads} \mid k \mbox{ heads})=P(k-1 \cap K)/P(K \mbox{ heads})=P(K-1)/P(K).$ I also was thinking about $P(A\mid A,B)=P(A\cap (A\cup B))/P(A\cup B)=P(A\cup (A\cap B))/P(A\cup B)=P(A)/(P(A)+P(B)-P(AB))$