Horse racing question probability Been thinking about this for a while.


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*Horse Campaign length: 10 starts

*Horse Runs this campaign: 5

*Horse will is guaranteed to win 1 in 10 this campaign


Question: what is the Probability of winning at the sixth start if it hasn't won in the first five runs?
20%? 
Thanks for looking at this.
I would love to how to calculate this.

I assume, that the horse will win only one race. And the probability of the (garanteed) win and a start at a race are equally like for every start/race.
The probability, that the horse has had no starts ($s_0$) is 
$\frac{{5 \choose 0}\cdot {5 \choose 5}}{{10 \choose 5}}$ and the probabilty that then the horse will win at the 6th start is 1 divided by the number of remaining starts. 
$P(w_6 \cap s_0)=P(s_0)\cdot P(w_6|s_0)=\frac{{5 \choose 0}\cdot {5 \choose 5}}{{10 \choose 5}}\cdot \frac{1}{5}$
The probability, that the horse has had one start ($s_1$) is 
$\frac{{5 \choose 1}\cdot {5 \choose 4}}{{10 \choose 5}}$ and the probabilty that then the horse will win at the 6th start is 1 divided by the number of remaining starts. 
$P(w_6 \cap s_1)=P(s_1)\cdot P(w_6|s_1)=\frac{{5 \choose 1}\cdot {5 \choose 4}}{{10 \choose 5}}\cdot \frac{1}{4}$
And so on ...
The probability of winning at the 6th start is $P(w_6)=P(w_6 \cap s_0)+P(w_6 \cap s_1)+P(w_6 \cap s_2)+P(w_6 \cap s_3)+P(w_6 \cap s_4)$
