Odd Number Probability How do I calculate the probability of randomly picking a number in the range from 0 to 10000 where exactly two of the digits is odd. For ex 5001 or 1010 etc.. and how would I do the same calculation for a range that starts with an odd number for ex 10000 to 20000?
 A: Note that we can e.g. treat $5000$ as $05000$ the condition that the left one has exactly 2 odd digits is equivalent to the condition that the right one has 2 odd digits, because 0 is even. 
The process of drawing a random number out of $\{0,1, ..., 10000\}$ is therefore equivalent to drawing once from a distribution that has probability $10000/10001$ for 0 and $1/10001$ for 1 and 4 times from a uniform distribution over $\{0, ..., 9\}$ and ordering these numbers.
We therefore look for the probability that exactly two of these 5 digits are odd. 
The probability for the first digit to be odd is $1/10001$ and for the other 4 digits the probability for being odd is $0.5$ and order doesnt matter, therefore the probability you are looking for is:
$$ 10000/10001\cdot0.5^4 \cdot {{4}\choose{2}}$$
If you get a 1 for the left digit you cannot get two odd Digits because the maximal number is 10000 If you get a 0 for the left digit you need two odd digits out of the remaining 4 which is the above term
