In an interview, I was once told that I should not consider probability when talking about option greeks since from a mathematical point of view greeks have nothing to do with probability. That is of course true, from a mathematical point of view, but the way greeks are taught always takes into account the probability reasoning.
For example, in this website Volatility it says this:
A rise in the implied volatility of a call will decrease the delta for an in-the-money option, because it has a greater chance of going out-of-the-money, whereas for an out-of-the-money option, a higher implied volatility will increase the delta, since it will have a greater probability of finishing in the money.
Again, it is taking into account probability to explain vega. I understand that because I was also taught greeks through probability reasoning. Therefore, how can I understand greeks functioning without taking into account the probability reasoning? How would you explain the relationship vega vs delta without considering "chances of ending in the money"?