A fair coin is tossed n times. Let E be the event that the first toss is a head, and let $\displaystyle F_k$ be the event that there are k heads in total. For what values of n, k are E and $\displaystyle F_k$ independent?
What do I use here?
A fair coin is tossed n times. Let E be the event that the first toss is a head, and let $\displaystyle F_k$ be the event that there are k heads in total. For what values of n, k are E and $\displaystyle F_k$ independent?
What do I use here?
Is it clear to you that $\displaystyle P(E) = 0.5$?
If $\displaystyle 0 < k \leqslant n$, is it true that $\displaystyle P\left( {F_k } \right) = {\binom{n}{k}}\left( {0.5} \right)^n $?
$\displaystyle 0 < k < n$ then $\displaystyle P\left( {E \cap F_k } \right) = P\left( {F_k |E} \right)P(E) = {\binom{n-1}{k-1}}\left( {0.5} \right)^n $.
Can you find the values such that $\displaystyle P\left( E \right) \cdot P\left( {F_k } \right) = P\left( {E \cap F_k } \right)$?