A slight variation on probability of winning a number of games
A competition involves winning a number of games consecutively. The probability of a player winning any game is $p$. A typical competition might involve $6$ rounds (but it could be more or less - typically in the range $4 - 7$) and so, in a $6$ round event, a player must win $6$ games consecutively to win the competition (and cant win more than this nor does he have to lose a game to end his 'run')
A player takes part in $N$ independent competitions over a period of time.
What are the chances of him winning a total of $m$ games in these $N$ competitions?
I reckon the probability of any particular combination of Wins and losses is $p^m(1-p)^N$ - ignoring the fact that they do not have to lose a game if they win the whole competition But I am having real trouble working out the number of such combinations for different $m$ and $N$. Help greatly appreciated
Eventually I am interested in questions such as What are the chances a player will win more than $x$ games; or at least $x$ games; or less than $x$ games.