Expected Value Distributions

Overview

Students will build a probability distribution for a random variable and use it to calculate expected value. The lesson connects theoretical probabilities to an expected outcome, using a multiple-choice guessing scenario appropriate for grade 12 probability decision-making.

Learning intentions

Students will be able to:

• construct a probability distribution for the number of correct answers when guessing a multiple-choice test
• compute expected value from a probability distribution
• interpret what “expected score” means in a grading context

Success criteria

• I can list all possible values of the random variable (number correct) and find their probabilities.
• I can verify probabilities add up to 1.
• I can compute the expected value using (E(X)=\sum xP(x)).
• I can explain in words what the expected score represents.

Curriculum links

• Using Probability to Make Decisions — probability distributions and expected value (CCSS.MATH.CONTENT.HSS-MD.A.3)
• Functions and modeling connections via exponential/probability interpretation as needed (CCSS.MATH.CONTENT.HSF-TF.A.3, CCSS.MATH.CONTENT.HSA-SSE.B.3c)
• Representing constraints by equations/inequalities for grading schemes (CCSS.MATH.CONTENT.HSA-CED.A.3)

Lesson structure (60 minutes)

1. 0–7 min · Hook (real grading context). Teacher shows: “5-question multiple-choice quiz, 4 choices each. If you guess, what score should you expect?” Students discuss quick predictions and what “expect” means.

2. 7–18 min · Direct teach: define the random variable. Teacher introduces (X=) number of correct answers out of 5, and reviews theoretical probability idea: each question has probability (1/4) of correct and (3/4) of incorrect when guessing. Students create a simple table for one question: (P(\text{correct})) and (P(\text{incorrect})).

3. 18–30 min · Build the probability distribution. Teacher guides distribution construction for (X\in{0,1,2,3,4,5}), using the binomial counting idea: (P(X=k)=\binom{5}{k}(1/4)^k(3/4)^{5-k}). Students fill in probabilities for each (k) on a worksheet, using a provided (\binom{5}{k}) row and a calculator.

4. 30–35 min · Checkpoint: probability sanity check. Teacher asks students to sum all (P(X=k)) values and compare to 1 (within rounding). Students correct any rounding/entry errors and re-sum.

5. 35–47 min · Compute expected value. Teacher models (E(X)=\sum xP(x)) and demonstrates multiplying each (k) by its probability, then summing. Students compute (E(X)) step-by-step and record the final expected number correct.

6. 47–55 min · Interpret in grading language. Teacher presents a grading scheme prompt (teacher chooses one for today):

• Scheme A: “1 point per correct answer, 0 for incorrect.” Then expected score equals expected value of (X). Students write 2–3 sentences explaining what the expected score means (not the most likely score, but the long-run average).

7. 55–60 min · Exit ticket (quick check). Teacher gives: “What is the probability of getting exactly 2 correct?” and “What does expected value mean?” Students answer independently; teacher reviews for misconceptions.

Resources

• Probability distribution worksheet with (\binom{5}{k}) values prefilled for (k=0) to (5)
• Calculator or phone calculators
• Teacher slide/board with formulas for (P(X=k)) and (E(X))
• Small number line or X-value list (0–5) for visual support
• Graph paper or table template for probability and multiplication row

Assessment

• Formative during distribution building: teacher checks one or two computed (P(X=k)) entries before students complete the rest.
• Formative at the sum-to-1 checkpoint: students explain what to do if the sum is not near 1.
• Exit ticket: verify correct probability for (k=2) and a clear definition/interpretation of expected value.

Differentiation

• For the learner with 3rd-grade reading level:
• Provide a simplified sentence-support sheet: “Expected means the average you get over many tries.”
• Use large-print, color-coded columns (X, Probability, Multiply, Add) and prefilled (\binom{5}{k}).
• Offer step cards for the exact order: (1) find (P(X=k)) (2) multiply by k (3) add totals.
• For learners at 9–10 grade reading comprehension with 10 grade math:
• Keep the same tasks but require written justification for the interpretation step: “Why expected value is not necessarily the most frequent outcome.”
• Optionally ask them to compute expected score under a second scheme if time allows (but still keep within the 60 minutes by focusing on interpretation).
• For all students:
• Use rounding guidance (e.g., round probabilities to 4 decimals at intermediate steps) to prevent arithmetic overwhelm.
• Model one full row example before independent work to reduce working-memory load.
• Presentation supports: allow use of a calculator and allow work with a partner, but keep an independent exit ticket requirement.