Question 113276
This is a relatively straight-forward counting problem, but you have to be careful not to count someone twice.


First thing is that there are 30 History majors, but 6 of them are dual History/Business majors, that means that there are only 30 - 6 = 24 students that are ONLY History majors.  Likewise, of the 45 Business majors, 6 of them are dual, so 45 - 6 = 39 are ONLY Business.

In summary:


24 Only History
39 Only Business
6 History/Business
25 Engineering


All of that totals to 94, but there are 100 students, so that means 6 students have no major at all.


The business majors in the random selection could be either from the Business only group or the dual History/Business group, so that is a total of 45, plus there are 25 Engineering majors, making a total of 70 possibilities out of the 100 students.  Therefore the desired probability is 0.70.


As an aid to having this make sense, consider this:  The only way a student from this group could NOT be either an Engineering or Business major is if the student were a History ONLY major or had not selected his major yet.  There are 24 History only and 6 no major students for a total of 30, so there is a 0.30 probability that the student is NOT either Engineering or Business, or 1 - the probability that the student is from either of those two groups.


Hope this helps
John