's

You will run into different opinions about that. Some would pronounce the 's as another syllable -ez; others will not and pronounce it the same as Charles.
https://www.usingenglish.com/forum/ask-teacher/4013-how-should-i-pronounce.html

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"Just 15 seconds away from officially being born, baby Nevaeh reached up and grabbed the doctor's finger."
More: Picture: Baby Holds Doctor's Hand During C-section - Health & Science - CBN News - Christian News 24-7 - CBN.com

Odessa Dawn said:

"Just 15 seconds away from officially being born, baby Nevaeh reached up and grabbed the doctor's finger."
More: Picture: Baby Holds Doctor's Hand During C-section - Health & Science - CBN News - Christian News 24-7 - CBN.com

After listening to this video, I think that the second common rule (Charles) has been used. Am I right?
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It is the pronunciation all of us would use of doctor's.

Odessa Dawn said:

"Just 15 seconds away from officially being born, baby Nevaeh reached up and grabbed the doctor's finger."
More: Picture: Baby Holds Doctor's Hand During C-section - Health & Science - CBN News - Christian News 24-7 - CBN.com

After listening to this video, I think that the second common rule (Charles) has been used. Am I right?
​

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No. The 's' sound is clearly audible in the video. The baby grabbed 'the doctor's finger'. Because 'doctor' does not end with an 's' sound, the possessive apostrophe s needs to be added. It is a different situation with 'Charles' because that name already ends with an s.

OD, your practice of providing single quotes from other threads, with no further context, is not really helpful. It's a little frustrating having to follow a link to another thread, and possibly read the whole thread, just to find out the point of your question.If your question is relevant to that other thread, ask it there.

Odessa Dawn said:

Thank you all. I asked about the sound of 's right here. Is it (s) or (-ez)?

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Neither. It was /z/, partially devoiced because of the following voiceless /f/.