Yes and No
See Katz v. United States, 389 U.S. 347, 351 (1967):
What a person knowingly exposes to the public, even in his home or office, is not a subject of Fourth Amendment protection. But what he seeks to preserve as private, even in an area accessible to the public, may be constitutionally protected.
The glass has clearly been "knowingly exposed" ...
Hard to Tell
The individual consented to a search by letting the officer come in, and consent searches are held to be reasonable. Katz was an unconsented invasion of privacy (picking up sound through the glass in a telelphone booth). The glass has been "knowingly exposed", and the glass is in plain sight (Horton v. California, 496 U.S. 128. "...
First, welcome to LSE, Josh. And congratulations on such a nice first question.
The answer to your question is No.
Inviting an officer into your house to talk about a robbery does not give the officer the right to search your house. Talking and searching involve two different amendments, and you must waive your rights under each explicitly. However, under ...
The difference is stylistic only. There are not two different substantive probate cause standards in the U.S. law of constitutional criminal procedure. "Probable cause to believe" is simply old fashioned wording.
I will try to locate some authority to this effect if I can.