Statistics Flashcards: Compare Strategies Using Expected Value

What this deck covers

This deck focuses on Compare Strategies Using Expected Value, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 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 $0.05$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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