Some people are right that it's all kind of complex, but you've explained it all kind of backwards, starting with your use of "larger aperture" vs. "higher aperture"
The lens' f-number is defined as the focal length divided by the entrance pupil diameter. Essentially, a numerically "lower" f-number (meaning a large entry pupil diameter, and/or a lower focal length) will focus a larger cone of light onto the sensor. This can mean a better picture in low light situations because the sensor doesn't need to increase its sensitivity, which increases noise, and also means a narrower depth of field (which could be good or bad, depending on its application).
F-number applies to all cameras, even cheap disposable ones. The secret is that fixed-focus cameras (e.g. disposables or most cheap cell phone cameras) have a numerically "high" f-number, like f/22. This makes their depth of field very deep so they can be focused on anything from infinity to maybe 5 feet, but they have poor low-light sensitivity.
That depends on more than just one number...
Given two identical sensors, and a properly-focused image, this could be true but isn't automatic.
