Probability of an odd number in 10/20 lotto Say you have a lotto game 10/20, which means that 10 balls are drawn from 20.
How can I calculate what are the odds that the lowest drawn number is odd (and also how can I calculate the odds if it's even)?
So a detailed explanation:
we have numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20
and the drawn numbers were for example 3, 5, 8, 11, 12, 13, 14, 15, 18 and 19
so, we see now that lowest number is 3 and he is an odd number. 
So, as stated above, can you help me in finding out how to calculate such probability?

The total number of outcomes is ${20 \choose 10}$. Now count the total number of favorable outcomes:


*

*outcomes with lowest element 1 : ${19 \choose 9}$ ;

*outcomes with lowest element 3 : ${17 \choose 9}$ ;

*outcomes with lowest element 5 : ${15 \choose 9}$ ;

*outcomes with lowest element 7 : ${13 \choose 9}$ ;

*outcomes with lowest element 9 : ${11 \choose 9}$ ;

*outcomes with lowest element 11 : ${9 \choose 9} = 1$ ;


So the probability is $$\sum_{k\in \{9, 11, 13, 15, 17, 19 \}} { {k \choose 9} \over {20 \choose 10}} = {30616 \over 46189} \simeq 0.662842.$$
