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?
 A: 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 $p<1$. It is possible that $(1-p)^{850}$ becomes $0$ ?
A: The probability that you win at least once is:
$$1-\left(1-\frac{1}{850}\right)^{850}$$
A: 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
