Odds of winning contest I enter a contest every week with $850$ others that I have a random chance of being selected as the winner.  There is only one winner.  If I play for $850$ weeks, should I have a $100\%$ chance of being a winner?
The probability of beeing selected at least once is One minus the probability of beeing never selected (converse probability)

$P(X\geq 1)=1-P(X=0)$,

where $X$ is the random variable of beeing selected in 850 weeks. It is binomial distributed:

$X\sim Bin(850,p)$

$P(X=0)=\binom{850}{0}\cdot p^0\cdot (1-p)^{850}=(1-p)^{850}$

p is the probability of beeing selected in one week.

$1-P(X=0)$ is $1$ if $P(X=0)=0$.

We can assume that $p&lt;1$. It is possible that $(1-p)^{850}$ becomes $0$ ?
The winning chances are independent of each other. So, for every match you have chance of 1 for 850 members. Even if you play for infinite times, there is no guarantee for 100% winning. So for 850 matches you have probability of winning as 


  1/(850)^ 850


The above case is probability for winning in all the 850 matches.

The probability for winning atleast once is binomial. It is 


  1 - ( losing all the times ).


Losing all the times is  (no. of players - 1) / no. of players raised to power of no. of matches played