E.1 Finding grades. In my statistics class, the final average is based on the following grading distribution:

Presentations Quizzes Midterm 1 Midterm 2 Midterm 3 Final Exam
5% 20% 15% 15% 15% 30%

Suppose that Bob has an 83 quiz average, a 100 participation average, and the following three midterm exam grades: 71, 85, 79. What is the lowest grade that Bob can get on the final exam and still finish with a final average of 80 (or higher) for the course?

E.2 Unusual dice. A six-sided die is manufactured so that it had two sides marked with the number six, and no side marked with the number one. The other sides are marked normally. What is the expected value of rolling this six-sided die? Assume that each side is equally likely to land on top.

E.3 Unfair coin. A novelty coin is weighted so that it lands heads 70% of the time and tails only 30% of the time.

  1. If you flip the coin 50 times, how many heads should you expect?
  2. If you flipped the coin 50 times and got 37 heads, what is the average number of heads per flip?
  3. Explain what expected value is, and why we should not be too shocked that the expected number of heads was not exactly the same as what actually happened when we got 37 heads.
  4. What does the law of large numbers say about the average number of heads per flip if we keep flipping this coin more and more times?