Expected Value and Decisions: Precalculus Math Test Answer Key – PDF Download

Precalculus Expected Value Test Answer Key: questions 1–5

Question 1: Distribution check (2 points)

1. \(k=1-0.15-0.35-0.20=0.30\).
2. \(P(X \ge 2)=0.35+0.20=0.55\).

Scoring: 1 point for \(k=0.30\) (sum of probabilities equals 1); 1 point for \(0.55\).

Question 2: Carnival spinner (2 points)

1. \(P(0)=\frac{3}{6}=\frac{1}{2}\), \(P(3)=\frac{2}{6}=\frac{1}{3}\), \(P(6)=\frac{1}{6}\).
2. \(E=0(\frac{1}{2})+3(\frac{1}{3})+6(\frac{1}{6})=0+1+1=2\). The expected prize is $2.

Scoring: 1 point for the three correct probabilities; 1 point for the sum of products and the answer 2.

Question 3: Red lights on the way to school (3 points)

1. \(P(0)=\frac{12}{80}=0.15\), \(P(1)=\frac{28}{80}=0.35\), \(P(2)=\frac{24}{80}=0.30\), \(P(3)=\frac{16}{80}=0.20\).
2. \(E=0(0.15)+1(0.35)+2(0.30)+3(0.20)=0.35+0.60+0.60=1.55\).
3. \(200 \times 1.55=310\) red lights.

Scoring: 1 point per part.

Question 4: Card game at the fair (3 points)

1. \(E=12(\frac{1}{10})+3(\frac{3}{10})+0(\frac{6}{10})=1.2+0.9=2.1\). The expected payout is $2.10.
2. \(2.10-2=0.10\). The expected net result is +$0.10.
3. Not fair, because the expected net result is not 0. It favors the player by $0.10 per game. The fair price is $2.10.

Scoring: 1 point per part.

Question 5: Phone insurance (3 points)

1. \(0.06(450)=27\). The expected loss is $27.
2. No. The insurance costs $35, which is $8 more than the expected loss.
3. \(p(450)=35\) gives \(p=\frac{35}{450}\approx 0.0778\), which is about \(7.8\%\).

Scoring: 1 point per part.

Precalculus Expected Value Test Answer Key: questions 6–8

Question 6: Two penalty kicks (3 points)

1. \(P(0)=0.4 \times 0.4=0.16\); \(P(1)=2(0.6)(0.4)=0.48\); \(P(2)=0.6 \times 0.6=0.36\). The total is \(1\).
2. \(E=0(0.16)+1(0.48)+2(0.36)=0.48+0.72=1.2\) goals.

Scoring: part (a) 1 point for finding the end probabilities, 1 point for combining the two ways to get one goal; part (b) 1 point.

Question 7: TV warranty (2 points)

1. \(0.10(350)=35\). The expected repair cost is $35.
2. The warranty costs $80, which is $45 more than the expected repair cost, so it is not worth buying by expected value.

Scoring: 1 point per part.

Question 8: Entry fee for a game (2 points)

1. \(E=8(0.25)+3(0.5)+0(0.25)=2+1.5=3.5\). The expected payout is $3.50.
2. Per game: \(3.5-4=-0.5\). For 100 games: \(100(-0.5)=-50\). A loss of about $50.

Scoring: 1 point per part.

Scoring summary

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