Yes we can prove it, based on the percentage of treifos we find in similar animals which we have no right to assume these animals are immune from.

You have the kiddushin example backwards.

But using the statistics like this doesn't work to make a vadai.

The calculations assume a random sample of cows in the herd and that the likelihood of a treifa in each one is independent from the likelihood of treifos in all the others. But that's not quite right.

As an illustration, imagine you have a group of 100 people in the US. Based on statistics of the whole US population, a random sample would be expected to be about 50 men and 50 women. About 2 of them would be Jewish. Does that mean that every time you have a group of 100 people you can assume 50 are men and 50 are women, two of whom are Jewish? Of course not.

1) Even random samples can have outliers.

2) Most groups aren't random samples of the population and there are other factors increasing or decreasing the likelihood that the population will share characteristics of a random person in the US.

If you walk into a shul, you'd expect the vast majority of the people there to be Jewish. And if it's the ezras noshim on shabbos morning, you'd expect the vast majority to be women. That's because these groups are not random samples of the US population. And in the ezras noshim scenario, it's not even a random sample of Jewish people.

This was, I think, part of the point

@moko was making about cows. The cows being milked do not have the same rate of treifos as cows being shechted because the milk cows are younger and healthier. So the shechted cows likely have more treifos than a stam random cow. Similarly, even the milk cows might have a below average rate of treifos because of the conditions they live in.