To measure the lifetime of a specific particle one needs to look at very many such particles in order to calculate the average. It cannot matter when the experimentalist actually starts his stopwatch to measure the time it takes for the particles to decay. If he measures them now or in 5 minutes makes no difference, since he still needs to take an average. If he measures later there will be particles out of the picture already (those who have decayed in the last 5 min), which won't contribute and the ones his measuring now behave (statistically) the very same, of course. 

I have just read the following in Introduction to Elementary Particles by Griffiths:


  Now, elementary particles have no memories, so the probability of a given muon
  decaying in the next microsecond is independent of how long ago that muon was
  created. (It's quite different in biological systems: an 80-year-old man is much more likely to die in the next year than is a 20-year-old, and his body shows the signs of eight decades of wear and tear, But all muons are identical, regardless of when they were produced; from an actuarial point of view they’re all on an equal footing.)


But this is not really the view I had. I was imagining, that a particle that has existed for a while is analogous to the 80 year old man, since it will probably die (decay) soon. It just doesn't matter because we are looking at a myriad of particles, so statistically there will be about as many old ones as babies. On the other hand it is true that I cannot see if a specific particle has already lived long or not; they are all indistinguishable. Still I am imagining particles as if they had an inner age, but one just can't tell by their looks. So is the view presented in Griffiths truer than mine or are maybe both valid?

How can one argue why my view is wrong?

It's impossible to say whether you are correct or Griffiths is correct a priori -- that is, before having any experience of how the world works.  You need to do experiments, and Griffiths' version agrees with experiments better than yours.

The basic experiment involves detecting the products of decayed particles.  Suppose we have some process that happens pretty quickly and produces a certain amount of some unstable particle.  The particles were all produced at basically the same time, so they all have basically the same "age".

Now, if we just measure how many decays happen per second, Griffiths' version predicts that you'll see an exponential falloff in that number -- as Griffiths does a fine job of explaining.  And that's what we actually see when we do experiments like this.

In your version, you would expect to see very few decays until some fixed time after the production, then a sudden rush of decays, and then the decays would mostly stop because the particles would basically all be gone.  But that's just not what we see.

Again, there's no reason the laws of the universe have to work the way Griffiths says, and not work the way you say.  It's just that the predictions of the two versions are testable, and only Griffiths' version agrees with experiments.  That's science!
