Compare Strategies Using Expected Value - Statistics
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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?
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?
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$\Delta P=p(d_B-d_A)=0.08\cdot 500=40$. Premium difference equals expected deductible difference.
$\Delta P=p(d_B-d_A)=0.08\cdot 500=40$. Premium difference equals expected deductible difference.
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What must be true about probabilities $p_i$ in an expected value model?
What must be true about probabilities $p_i$ in an expected value model?
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$\sum p_i=1$ and each $0\le p_i\le 1$. Probabilities must sum to 1 and each be between 0 and 1.
$\sum p_i=1$ and each $0\le p_i\le 1$. Probabilities must sum to 1 and each be between 0 and 1.
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Choose the lower expected cost: Policy A $E=1500$ or Policy B $E=1475$.
Choose the lower expected cost: Policy A $E=1500$ or Policy B $E=1475$.
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Policy B. Choose lower expected cost: $1475 < 1500$.
Policy B. Choose lower expected cost: $1475 < 1500$.
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Find $E(\text{total})$ if $P=800$, minor: $p=0.10$ cost $200$, major: $p=0.02$ cost $1000$, else $0$.
Find $E(\text{total})$ if $P=800$, minor: $p=0.10$ cost $200$, major: $p=0.02$ cost $1000$, else $0$.
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$800+0.10\cdot 200+0.02\cdot 1000=840$. Premium plus expected costs from both accident types.
$800+0.10\cdot 200+0.02\cdot 1000=840$. Premium plus expected costs from both accident types.
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Find the expected total cost: premium $P=1000$, accident prob $0.05$, deductible $d=500$ (else $0$).
Find the expected total cost: premium $P=1000$, accident prob $0.05$, deductible $d=500$ (else $0$).
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$1000+0.05\cdot 500=1025$. Premium plus expected deductible: $1000 + 0.05(500)$.
$1000+0.05\cdot 500=1025$. Premium plus expected deductible: $1000 + 0.05(500)$.
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Find $p$ where two options tie: A total $=1000+200p$, B total $=900+400p$.
Find $p$ where two options tie: A total $=1000+200p$, B total $=900+400p$.
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$p=0.5$. Set costs equal: $1000 + 200p = 900 + 400p$, solve for $p$.
$p=0.5$. Set costs equal: $1000 + 200p = 900 + 400p$, solve for $p$.
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Which option is better for minimizing expected cost when $p=0.6$: A $=1000+200p$ or B $=900+400p$?
Which option is better for minimizing expected cost when $p=0.6$: A $=1000+200p$ or B $=900+400p$?
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Option A. At $p = 0.6$: A costs $1120$, B costs $1140$.
Option A. At $p = 0.6$: A costs $1120$, B costs $1140$.
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Which statement correctly compares strategies using expected value when outcomes are monetary costs?
Which statement correctly compares strategies using expected value when outcomes are monetary costs?
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Prefer smaller expected cost, even if outcomes vary. Expected value analysis prioritizes average cost over variability.
Prefer smaller expected cost, even if outcomes vary. Expected value analysis prioritizes average cost over variability.
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Find $E(X)$ if $X=200$ with probability $0.1$ and $X=2000$ with probability $0.02$ and $0$ otherwise.
Find $E(X)$ if $X=200$ with probability $0.1$ and $X=2000$ with probability $0.02$ and $0$ otherwise.
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$E(X)=60$. Calculate $0.1(200) + 0.02(2000) = 20 + 40 = 60$.
$E(X)=60$. Calculate $0.1(200) + 0.02(2000) = 20 + 40 = 60$.
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What is the expected cost formula for a policy with premium $P$ and random out-of-pocket cost $X$?
What is the expected cost formula for a policy with premium $P$ and random out-of-pocket cost $X$?
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$E(\text{total})=P+E(X)$. Total expected cost is premium plus expected out-of-pocket.
$E(\text{total})=P+E(X)$. Total expected cost is premium plus expected out-of-pocket.
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Which option minimizes expected cost: choose the strategy with larger or smaller expected value?
Which option minimizes expected cost: choose the strategy with larger or smaller expected value?
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Choose the strategy with smaller $E(\text{cost})$. Lower expected cost means better financial outcome.
Choose the strategy with smaller $E(\text{cost})$. Lower expected cost means better financial outcome.
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Find $E(X)$ if $X=0$ with probability $0.7$ and $X=500$ with probability $0.3$.
Find $E(X)$ if $X=0$ with probability $0.7$ and $X=500$ with probability $0.3$.
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$E(X)=150$. Calculate $0.7(0) + 0.3(500) = 150$.
$E(X)=150$. Calculate $0.7(0) + 0.3(500) = 150$.
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Identify the expected out-of-pocket cost if you pay $d$ with probability $p$ and $0$ otherwise.
Identify the expected out-of-pocket cost if you pay $d$ with probability $p$ and $0$ otherwise.
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$E= pd$. Multiply deductible by probability of paying it.
$E= pd$. Multiply deductible by probability of paying it.
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What is the expected value of a Bernoulli loss: pay $L$ with probability $p$, else pay $0$?
What is the expected value of a Bernoulli loss: pay $L$ with probability $p$, else pay $0$?
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$E= pL$. Expected value is probability times loss amount.
$E= pL$. Expected value is probability times loss amount.
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Correctly compute and choose: Low ded $P=1200$, costs $100$ (minor), $500$ (major); High ded $P=900$, costs $400$ (minor), $1500$ (major); $p_m=0.10,p_M=0.02$.
Correctly compute and choose: Low ded $P=1200$, costs $100$ (minor), $500$ (major); High ded $P=900$, costs $400$ (minor), $1500$ (major); $p_m=0.10,p_M=0.02$.
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High deductible, since $E_L=1220$ and $E_H=970$. Low: $1200 + 0.1(100) + 0.02(500) = 1220$; High: $900 + 0.1(400) + 0.02(1500) = 970$.
High deductible, since $E_L=1220$ and $E_H=970$. Low: $1200 + 0.1(100) + 0.02(500) = 1220$; High: $900 + 0.1(400) + 0.02(1500) = 970$.
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What is the expected value formula for outcomes $x_i$ with probabilities $p_i$?
What is the expected value formula for outcomes $x_i$ with probabilities $p_i$?
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$E(X)=\sum p_i x_i$. Multiply each outcome by its probability and sum.
$E(X)=\sum p_i x_i$. Multiply each outcome by its probability and sum.
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Identify the break-even major-accident probability $p$ if low ded costs $500$ more in premium but saves $2500$ in major loss.
Identify the break-even major-accident probability $p$ if low ded costs $500$ more in premium but saves $2500$ in major loss.
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$p=\frac{500}{2500}=0.2$. Break-even when premium savings equals expected loss difference.
$p=\frac{500}{2500}=0.2$. Break-even when premium savings equals expected loss difference.
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Identify the expected value of a mixed strategy: choose policy A with probability $q$ and policy B with probability $1-q$.
Identify the expected value of a mixed strategy: choose policy A with probability $q$ and policy B with probability $1-q$.
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$E=qE_A+(1-q)E_B$. Weighted average of expected values from each policy.
$E=qE_A+(1-q)E_B$. Weighted average of expected values from each policy.
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What is the expected total annual cost of an insurance plan with premium $P$ and random out-of-pocket $X$?
What is the expected total annual cost of an insurance plan with premium $P$ and random out-of-pocket $X$?
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$E(\text{total})=P+E(X)$. Add fixed premium to expected variable costs.
$E(\text{total})=P+E(X)$. Add fixed premium to expected variable costs.
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Find $E(X)$ if $X=0$ with probability $0.7$ and $X=500$ with probability $0.3$.
Find $E(X)$ if $X=0$ with probability $0.7$ and $X=500$ with probability $0.3$.
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$E(X)=150$. $0(0.7) + 500(0.3) = 150$.
$E(X)=150$. $0(0.7) + 500(0.3) = 150$.
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Find $E(X)$ if $X=200$ with probability $0.1$ and $X=2000$ with probability $0.02$ and $0$ otherwise.
Find $E(X)$ if $X=200$ with probability $0.1$ and $X=2000$ with probability $0.02$ and $0$ otherwise.
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$E(X)=60$. $200(0.1) + 2000(0.02) + 0(0.88) = 20 + 40 = 60$.
$E(X)=60$. $200(0.1) + 2000(0.02) + 0(0.88) = 20 + 40 = 60$.
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Compute the expected out-of-pocket cost with deductible $d=500$ for loss $L=300$.
Compute the expected out-of-pocket cost with deductible $d=500$ for loss $L=300$.
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$0$. Loss $300 < 500$ deductible, so pay nothing.
$0$. Loss $300 < 500$ deductible, so pay nothing.
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Compute the expected out-of-pocket cost with deductible $d=500$ for loss $L=1200$.
Compute the expected out-of-pocket cost with deductible $d=500$ for loss $L=1200$.
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$700$. Loss $1200 - 500$ deductible = $700$ out-of-pocket.
$700$. Loss $1200 - 500$ deductible = $700$ out-of-pocket.
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Compute expected total cost: premium $P=900$ and expected out-of-pocket $E(X)=150$.
Compute expected total cost: premium $P=900$ and expected out-of-pocket $E(X)=150$.
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$1050$. $900 + 150 = 1050$ total expected cost.
$1050$. $900 + 150 = 1050$ total expected cost.
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State the formula for expected value $E(X)$ for a discrete random variable.
State the formula for expected value $E(X)$ for a discrete random variable.
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$E(X)=\sum x,P(X=x)$. Sum each outcome times its probability.
$E(X)=\sum x,P(X=x)$. Sum each outcome times its probability.
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Which plan is cheaper in expectation? A: $P=600,d=1000$; B: $P=900,d=250$; loss is $L=2000$ with prob $0.1$.
Which plan is cheaper in expectation? A: $P=600,d=1000$; B: $P=900,d=250$; loss is $L=2000$ with prob $0.1$.
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Plan A. A: $600+0.1(1000)=700$; B: $900+0.1(1750)=1075$.
Plan A. A: $600+0.1(1000)=700$; B: $900+0.1(1750)=1075$.
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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 $0.05$.
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 $0.05$.
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Plan A. A: $700+0.2(0)+0.05(3000)=850$; B: $1000+0.2(350)+0.05(3750)=1257.5$.
Plan A. A: $700+0.2(0)+0.05(3000)=850$; B: $1000+0.2(350)+0.05(3750)=1257.5$.
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Find $E(X)$ for $X=100$ with probability $0.4$, $X=300$ with probability $0.1$, and $0$ otherwise.
Find $E(X)$ for $X=100$ with probability $0.4$, $X=300$ with probability $0.1$, and $0$ otherwise.
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$E(X)=70$. $100(0.4) + 300(0.1) + 0(0.5) = 40 + 30 = 70$.
$E(X)=70$. $100(0.4) + 300(0.1) + 0(0.5) = 40 + 30 = 70$.
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Identify the correct expected value if probabilities are $p_1,p_2,p_3$ for costs $c_1,c_2,c_3$ and $p_1+p_2+p_3=1$.
Identify the correct expected value if probabilities are $p_1,p_2,p_3$ for costs $c_1,c_2,c_3$ and $p_1+p_2+p_3=1$.
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$E= p_1c_1+p_2c_2+p_3c_3$. Sum of probability-weighted costs for three outcomes.
$E= p_1c_1+p_2c_2+p_3c_3$. Sum of probability-weighted costs for three outcomes.
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Which plan is cheaper in expectation? A: $P=400,E(X)=300$; B: $P=650,E(X)=80$.
Which plan is cheaper in expectation? A: $P=400,E(X)=300$; B: $P=650,E(X)=80$.
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Plan A. A: $400+300=700$; B: $650+80=730$; A is cheaper.
Plan A. A: $400+300=700$; B: $650+80=730$; A is cheaper.
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