Does increasing the dice pool make messy criticals more likely?

I just read the mechanics for dice rolling in Vampire: the Masquerade 5th edition and something is bothering me. It seems to me that the larger your dicepool is, the more likely it is to get a messy critical.

Just in case somebody wants to answer without knowing the rules, here is a very brief rundown:

A character rolls a number of d10s called a dicepool. For the vast majority of rolls, at least one of those dice would be a hunger die. Let’s just assume it’s a single one for simplicity. If at least two dice come up as 10s, that’s considered a critical. If at least one of those 10s is on a hunger die, then it’s a messy critical where the character succeeds spectacularly but in the most direct and brutal way possible. Picking a lock with a messy critical can lead to the character ripping the door off the hinges – grants passage but it’s not subtle.

These are the relevant rules here. It seems to me that the larger the dicepool is, the more of a chance for a messy critical. My intuition is the following:

  • with a dicepool of 3, if the hunger die comes up at 10, then you have 2 chances to roll a 10 on the other dice.
  • with a dicepool of 7, if the hunger die comes up at 10, then you have 6 chances to roll a 10 on the other dice.

Is my intuition here correct? Is a master at picking locks would be more likely to let the Beast do his job than somebody who’s just average at locks? I am not sure how to properly calculate the odds of messy criticals.