Back to Exercise: Evaluate and compare strategies using expected values

Exercises: Evaluate and Compare Strategies Using Expected Values

Work through each section in order. For every comparison, FIRST state the
direction: an expected COST means lower is better; an expected PAYOFF means
higher is better. Model each strategy as its own distribution, compute its
expected value, then compare the totals. For insurance problems, remember
expected cost = premium + expected out-of-pocket. Show your arithmetic.

Grade 11·22 problems·~35 min·Common Core Math - HS Statistics and Probability·standard·hss-md-b-5b

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Warm-Up: Direction and Single-Option Expected Value

These problems review computing one expected value and stating the cost/payoff direction.

You are comparing two strategies by their expected annual cost (in dollars). Strategy X has an expected cost of $\text{\char"0024}950$ and Strategy Y has an expected cost of $\text{\char"0024}1{,}237.50$ . Which strategy should you recommend, and why?

Strategy X, because for an expected COST, lower is better.

Strategy Y, because a higher expected value is always better.

Strategy Y, because it costs more, so it must offer more.

Neither — expected costs cannot be compared directly.

A prize wheel costs $\text{\char"0024}5$ to spin. It pays $\text{\char"0024}20$ with probability $0.3$ and $\text{\char"0024}0$ with probability $0.7$ . Model this as a NET payoff (winnings minus the $\text{\char"0024}5$ cost). What is the expected net payoff, in dollars? Enter a number (use a negative sign if it is a loss).

To compare two competing strategies, you model each one as its own ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ distribution, compute each strategy's ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , and then compare those values. (First blank: "payoff or cost". Second and third blanks: the two-word name of the average you compute.)

Fluency Practice

Compute each strategy's expected value and compare. State the direction.

An auto policy has a $\text{\char"0024}1{,}200$ annual premium and a $\text{\char"0024}250$ deductible paid per accident. For this driver the chance of having at least one accident (which triggers the full deductible) is $0.15$ . The expected annual cost is the premium plus the expected out-of-pocket payment. Compute the expected annual cost, in dollars.

Two prize wheels each cost the same to play. Wheel P has an expected PAYOFF of $\text{\char"0024}8$ . Wheel Q has an expected PAYOFF of $\text{\char"0024}11$ . Which wheel should you choose to maximize winnings, and why?

Wheel Q, because for an expected PAYOFF, higher is better.

Wheel P, because for any expected value, lower is better.

Wheel P, because a smaller payoff is safer.

Either — equal cost means the payoffs do not matter.

Policy B has an $\text{\char"0024}800$ annual premium and a $\text{\char"0024}1{,}000$ deductible paid per accident. For a driver whose chance of an accident is $0.15$ , compute the expected annual cost, in dollars.

For a given driver, Policy A has an expected annual cost of $\text{\char"0024}1{,}237.50$ and Policy B has an expected annual cost of $\text{\char"0024}950$ . Which policy do you recommend?

Policy B, because $\text{\char"0024}950 < \text{\char"0024}1{,}237.50$ and lower expected cost is better.

Policy A, because it has a smaller deductible.

Policy A, because a higher cost gives better coverage.

Cannot decide without knowing each premium.

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