Compare Strategies Using Expected Value - Statistics and Probability

1. Compute the expected out-of-pocket cost with deductible $d=500$ for loss $L=1200$.

   $700$. Loss $1200 - 500$ deductible = $700$ out-of-pocket.

2. Which plan is cheaper in expectation? A: $P=400,E(X)=300$; B: $P=650,E(X)=80$.

   Plan A. A: $400+300=700$; B: $650+80=730$; A is cheaper.

3. Identify the break-even major-accident probability $p$ if low ded costs $500$ more in premium but saves $2500$ in major loss.

   $p=\frac{500}{2500}=0.2$. Break-even when premium savings equals expected loss difference.

4. What is the expected value formula for outcomes $x_i$ with probabilities $p_i$?

   $E(X)=\sum p_i x_i$. Multiply each outcome by its probability and sum.

5. Which option minimizes expected cost: choose the plan with larger or smaller expected total cost?

   Choose the plan with the smaller expected total cost. Lower expected total cost means less money spent on average.

6. What is the expected cost formula for a policy with premium $P$ and random out-of-pocket cost $X$?

   $E(\text{total})=P+E(X)$. Total expected cost is premium plus expected out-of-pocket.

7. Find the break-even premium difference: Policy A has deductible $d_A=500$, Policy B has $d_B=1000$, accident prob $p=0.08$. What premium advantage makes B equal A?

   $\Delta P=p(d_B-d_A)=0.08\cdot 500=40$. Premium difference equals expected deductible difference.

8. Which option minimizes expected cost: choose the strategy with larger or smaller expected value?

   Choose the strategy with smaller $E(\text{cost})$. Lower expected cost means better financial outcome.

9. Identify the correct expected value if probabilities are $p_1,p_2,p_3$ for costs $c_1,c_2,c_3$ and $p_1+p_2+p_3=1$.

   $E= p_1c_1+p_2c_2+p_3c_3$. Sum of probability-weighted costs for three outcomes.

10. What is the expected value of a Bernoulli loss: pay $L$ with probability $p$, else pay $0$?

    $E= pL$. Expected value is probability times loss amount.

11. Which plan is cheaper in expectation? A: $P=600,d=1000$; B: $P=900,d=250$; loss is $L=2000$ with prob $0.1$.

    Plan A. A: $600+0.1(1000)=700$; B: $900+0.1(1750)=1075$.

12. Find $E(X)$ if $X=200$ with probability $0.1$ and $X=2000$ with probability $0.02$ and $0$ otherwise.

    $E(X)=60$. $200(0.1) + 2000(0.02) + 0(0.88) = 20 + 40 = 60$.

13. Which statement correctly compares strategies using expected value when outcomes are monetary costs?

    Prefer smaller expected cost, even if outcomes vary. Expected value analysis prioritizes average cost over variability.

14. Find $E(X)$ if $X=0$ with probability $0.7$ and $X=500$ with probability $0.3$.

    $E(X)=150$. Calculate $0.7(0) + 0.3(500) = 150$.

15. Identify the expected value of a constant cost $X=k$ (probability $1$).

    $E(X)=k$. Constant value has expected value equal to itself.

16. What is the expected value of a cost with outcomes $c_1,c_2$ and probabilities $p,1-p$?

    $E=pc_1+(1-p)c_2$. Weighted average of two cost outcomes.

17. What must be true about probabilities $p_i$ in an expected value model?

    $\sum p_i=1$ and each $0\le p_i\le 1$. Probabilities must sum to 1 and each be between 0 and 1.

18. Compute expected total cost: premium $P=900$ and expected out-of-pocket $E(X)=150$.

    $1050$. $900 + 150 = 1050$ total expected cost.

19. Find $E(X)$ if $X=200$ with probability $0.1$ and $X=2000$ with probability $0.02$ and $0$ otherwise.

    $E(X)=60$. Calculate $0.1(200) + 0.02(2000) = 20 + 40 = 60$.

20. Find the expected total cost: premium $P=1000$, accident prob $0.05$, deductible $d=500$ (else $0$).

    $1000+0.05\cdot 500=1025$. Premium plus expected deductible: $1000 + 0.05(500)$.

21. Which plan has smaller expected total cost? A: $P=700,d=1000$; B: $P=1000,d=250$; minor $L=600$ prob $0.2$, major $L=4000$ prob $0.05$.

    Plan A. A: $700+0.2(0)+0.05(3000)=850$; B: $1000+0.2(350)+0.05(3750)=1257.5$.

22. Correctly compute and choose: Low ded $P=1200$, costs $100$ (minor), $500$ (major); High ded $P=900$, costs $400$ (minor), $1500$ (major); $p_m=0.10,p_M=0.02$.

    High deductible, since $E_L=1220$ and $E_H=970$. Low: $1200 + 0.1(100) + 0.02(500) = 1220$; High: $900 + 0.1(400) + 0.02(1500) = 970$.

23. Identify the expected out-of-pocket cost if you pay $d$ with probability $p$ and $0$ otherwise.

    $E= pd$. Multiply deductible by probability of paying it.

24. Find $E(X)$ if $X=0$ with probability $0.7$ and $X=500$ with probability $0.3$.

    $E(X)=150$. $0(0.7) + 500(0.3) = 150$.

25. Find $E(X)$ for $X=100$ with probability $0.4$, $X=300$ with probability $0.1$, and $0$ otherwise.

    $E(X)=70$. $100(0.4) + 300(0.1) + 0(0.5) = 40 + 30 = 70$.

26. Find $p$ where two options tie: A total $=1000+200p$, B total $=900+400p$.

    $p=0.5$. Set costs equal: $1000 + 200p = 900 + 400p$, solve for $p$.

27. Which option is better for minimizing expected cost when $p=0.6$: A $=1000+200p$ or B $=900+400p$?

    Option A. At $p = 0.6$: A costs $1120$, B costs $1140$.

28. What is the expected total annual cost of an insurance plan with premium $P$ and random out-of-pocket $X$?

    $E(\text{total})=P+E(X)$. Add fixed premium to expected variable costs.

29. Identify the expected value of a mixed strategy: choose policy A with probability $q$ and policy B with probability $1-q$.

    $E=qE_A+(1-q)E_B$. Weighted average of expected values from each policy.

30. Choose the lower expected cost: Policy A $E=1500$ or Policy B $E=1475$.

    Policy B. Choose lower expected cost: $1475 < 1500$.

31. Find $E(\text{total})$ if $P=800$, minor: $p=0.10$ cost $200$, major: $p=0.02$ cost $1000$, else $0$.

    $800+0.10\cdot 200+0.02\cdot 1000=840$. Premium plus expected costs from both accident types.

32. State the formula for expected value $E(X)$ for a discrete random variable.

    $E(X)=\sum x\,P(X=x)$. Sum each outcome times its probability.

33. Compute the expected out-of-pocket cost with deductible $d=500$ for loss $L=300$.

    $0$. Loss $300 < 500$ deductible, so pay nothing.

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