9 Expected Value

9.1 An Example: GPA

Question: If a student receives 5 A’s, 4 B’s, and 1 C, what is the student’s GPA? (Assume all courses are equally weighted and that an A is worth 4, a B with 3, and a C worth 2.)

  1. Explain why \(\frac{4 + 3 + 2}{3}\) is not the correct answer.

  2. Show how to correctly calculate the GPA by listing the 10 scores individually.

  3. Factor your expression into the following form (fill in the missing numerators):

\[ \mbox{GPA} = 4 \cdot \frac{\phantom{5}}{10} + 3 \cdot \frac{\phantom{4}}{10} + 2 \cdot \frac{\phantom{1}}{10} \]

  1. Let \(X\) be the random variable that results from randomly selecting a course and recording its grade (on a 4 point scale). Create a probability table for \(X\).

  2. How are the numbers in your expression for GPA in #3 related to the numbers in your probability table?

9.2 Generalizing

We can generalize this idea to compute the mean (more commonly called expected value) of any random variable \(X\). This is denoted either \(\operatorname{E}(X)\) or \(\mu_X\).

  1. Let \(X\) be defined by the probability table below. Compute \(\operatorname{E}(X)\).
value of \(X\) 0 1 2 3
probability 0.2 0.3 0.4 0.1

  1. Let \(H\) be the number of heads in 3 tosses of a fair coin.

    1. Create a probability table for \(H\)
    2. Compute \(E(H)\).

  2. Use good mathematical notation to write down the definition:

\[ \operatorname{E}(X) = \phantom{\sum k p(X = k)} \]

  1. A raffle has 1000 tickets. Holders of 4 of the tickets get a prize. The other 996 are worth nothing. The four prizes are worth $500, $200, $50, and $50. Let \(V\) be the value of a random raffle ticket.

    1. Create a probabilty table for \(V\).
    2. Compute \(\operatorname{E}(V)\).
    3. What does \(\operatorname{E}(V)\) tell us about the raffle tickets?

  1. Let \(D\) be the absolute value of the difference between the values of two 4-sided dice.

    1. Create a probability table for \(D\).
    2. Compute \(\operatorname{E}(D)\).
    3. What is \(p(D = \operatorname{E}(D))\)?
    4. What is \(p(D < \operatorname{E}(D))\)?
    5. What is \(p(D > \operatorname{E}(D))\)?

We can do these already now, but there are easier ways that take advantage of properties of expected value that we haven’t learned yet.

  1. In a hand of 5 cards from a standard deck, what is the expected number of diamonds?

  2. If you roll 5 standard dice, what is the expected number of 6’s?

These two are a bit more challenging, but still doable.

  1. In a hand of 5 cards from a standard deck, what is the expected number of suits?

  2. If you roll 5 standard dice, what is the expected number of unique numbers rolled?