Final answer:
An A grade is not possible with the existing grades even if a 100 is scored on the final exam. To earn a B, the student must score at least a 68 on the final exam. The probability of a course having a final exam or a research paper is 0.86, and the probability of a course having neither is 0.14.
Step-by-step explanation:
To determine if an A grade is possible with a 100 on the final exam, we calculate the average including the final exam grade. The grades are 88, 78, 86, and a hypothetical 100. The average is calculated as follows: (88+78+86+100)/4 = 352/4 = 88. Since an average of 90 is required for an A, this means an A is not possible even with a perfect score on the final.
For a B grade, which requires a final average of at least 80, we use the equation of the form (88+78+86+x)/4 = 80, where x represents the final exam score. Solving for x, we find x = (80*4) - (88+78+86), so x = 320 - 252 = 68. The student must get at least a 68 on the final to earn a B in the course.
The probability question related to final exams and research papers can be solved using the principles of probability. For the first part, the probability that a course has a final exam or requires a research paper is given by P(F or R) = P(F) + P(R) - P(F and R), which equals 0.72 + 0.46 - 0.32 = 0.86. For the second part, the probability that a course has neither requirement is 1 - P(F or R), which equals 1 - 0.86 = 0.14.