Likelihood of HIV infection/ transmission if infected blood comes into contact with finger cut?

It all depends on several factors;
for one if it was a finger cut that was an older wound and healing the likelihood of transmission is low.
If the finger cut was fresh your transmission risk is high and still it would depend on several factors,
at what stage was the person with HIV, if they were on a drug regime, how much virus was in the blood, how much blood you came in contact with, and if the blood was transmitted through an artery or vein.
Just to relieve your anxiety, the odds of contracting HIV through a needle stick injury are 0.3%,
relax, and go get tested 3 months, 6 months, 1 yr....

About 95%. Go get tested now, in 3 months, 6 months, and 1 year. If after a year your tests come back negative, then you are OK.

Realize that a fresh cut is PUSHING blood out. If you've got raw skin exposed, then you've got a problem.

Unfortunately, my friend, there is a big possibility for it.

Very likely if not treated properly.