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"Second conditional (used to talk about unlikely, imaginary or untrue events or situations)If + past – would + infinitive"
More: Conditional Sentences - English Grammar & Usage
Then we have:
"Then someone suggested that if I had Oxford University on my CV I might improve my chances in law. So I applied to do a Masters at Oxford."
More: Postgraduate Lives
Does this mean that might can be used in informal English instead of would?
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The writer is studying for a masters, having already a first-class degree BA from Durham (another very good university). Her writing (and her thinking) are very careful. What she means is that people suggested 'If you have Oxford on your CV it will increase your chances....'. (If there are 'chances', there must be 'a chance that not...', so there is no need to say it might happen.)
This degree of distancing from certainty about a possible outcome is conventional among lawyers - her chosen field.