Why did people expect the number of electrons emitted because of the photoelectric effect to rise with the radiation's intensity? The number of electrons emitted because of the photoelectric effect 


*

*rises with the frequency of the radiation

*is not influenced by the intensity of the radiation (As Chad points out, this is wrong, the electron's energy isn't influenced by the radiation's intensity.)


Why was this not expected when it was discovered?
 A: You're mistaken in your history. The number of electrons emitted does, in fact, increase with the intensity of the radiation. Higher intensity means more photons, which means more chances to knock an electron loose.
It's the energy of the emitted electrons that people expected to increase that doesn't. In a classical wave picture, an increased intensity means a higher amplitude of the wave, which means more dramatic "shaking" of the electrons, and should thus lead to electrons being ejected with higher energy. In the quantum model, the emission is caused by a single photon, and the energy is determined by the frequency of that photon. Thus, the energy depends on the frequency but not the intensity.
A: The energy of a photon depends on the frequency: $E=h \nu$. This is not a classical result.
So classically: To extract electrons from a material you need a minimal energy. This can be provided by a minimal intensity.
But: from a quantum mechanical point of view this reasoning breaks down. That's the discovery of the photoelectric effect (and its correct explanation): to extract an electron you need a minimal energy that can be provided by photons above a frequency (and thus energy) threshold.
If the energy of the incident photon is higher than the "extraction work" of the material (the energy needed to extract the electron) this additional energy is transformed in kinetic energy of the [free] electron.
Therefore an increase in the intensity of light whose frequency is bellow the threshold cannot help as in the classical case.
Increasing the intensity of the light means a higher number of photon (thus more total power but I does not mean more energy per photon), this can allow to extract more electrons ("one by one").
A: Classically, light was thought to only have the characteristics of wave. The energy of a wave depends solely on the intensity not frequency. To knock out an electron one expects supplying enough energy would make it escape from the nucleon's attraction, so people won't expect frequency affects the photoelectric effect.
