How can I calculate common digits in daily numbers? We have daily numbers with 4 digits and 3 digits. These winning numbers often,but not always, have 1 or more digits the same. It would seem there should be a strategy to calculate the percent of time the daily numbers have common digits.
It would seem to be true for any two numbers given the number of digits in each number. The number of digits in the 2 "daily numbers" can be the same or different.
A program can be written to count them. A general formula would be more satisfying. It does not appear to be that simple though because you can have numbers with multiple digits that are the same.

The probability of having a $d$ digit number with no repeated digits is:
$$
p_{nr}(d) = \frac{\binom{10}{d}}{10^d}.
$$
Thus, the probability of having a digit with any repetition is:
$$
p_r(d) = 1 - \frac{\binom{10}{d}}{10^d}.
$$
For $d=3$ we would have $p_r(3) = 0.28$ and for $d=4$ we have $p_r(4) = 0.496$.
