If Both Parents Have The O Blood Type Can An Off-spring Have B Positive Blood Type?

If mother is O + blood type and dad is O - blood type then what will be the blood group of their child?

I am afraid that is not quite possible. You must have heard of the terms 'dominant' and 'recessive'. As it goes, every trait of our body is determined by the combination of two alleles. Alleles can be dominant, that is they will show up in the trait whenever they are present; or they can be recessive. This means that they will only express themselves when both are recessive and there is not any dominant allele present to express itself.
When talking of blood groups, A and B are dominant and O is recessive. So, we can get an A blood group if the genetic make up is AA or AO. One allele is from each parent. But you can only get an O group if both parents are O blood group because:

 
  Mom   Dad

Blood group   O   O

Alleles   O   O   O   O

Children   OO  OO  OO OO

All children have blood group O.

If however, even one parent had a B in his blood group (blood group AB, OB, BB) then the offspring might have the blood group B. Let's take OB:

 
  Mom   Dad

Blood group   O   B

Alleles   O   O   O   B

Children   OO  OB  OO OO

You can see now that there is a one in four chance for a child to develop blood group B in this case.

If I have b+ blood and the father is either a or b blood type, can we have a baby with o+?