Worked solution P46-Easy-1. The probability is 0.23040; the expected trial of first success is 2.778. $$P(X=2)=(1-0.36{)}^1(0.36)=0.23040.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-2. The probability is 0.12979; the expected trial of first success is 2.083. $$P(X=3)=(1-0.48{)}^2(0.48)=0.12979.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-3. The probability is 0.07243; the expected trial of first success is 2.174. $$P(X=4)=(1-0.46{)}^3(0.46)=0.07243.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-4. The probability is 0.05400; the expected trial of first success is 2.564. $$P(X=5)=(1-0.39{)}^4(0.39)=0.05400.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-5. The probability is 0.01216; the expected trial of first success is 1.887. $$P(X=6)=(1-0.53{)}^5(0.53)=0.01216.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-6. The probability is 0.04449; the expected trial of first success is 4.000. $$P(X=7)=(1-0.25{)}^6(0.25)=0.04449.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-7. The probability is 0.22110; the expected trial of first success is 3.030. $$P(X=2)=(1-0.33{)}^1(0.33)=0.22110.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-8. The probability is 0.14814; the expected trial of first success is 3.030. $$P(X=3)=(1-0.33{)}^2(0.33)=0.14814.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-9. The probability is 0.10290; the expected trial of first success is 3.333. $$P(X=4)=(1-0.30{)}^3(0.30)=0.10290.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-10. The probability is 0.03510; the expected trial of first success is 2.083. $$P(X=5)=(1-0.48{)}^4(0.48)=0.03510.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-11. The probability is 0.04258; the expected trial of first success is 2.941. $$P(X=6)=(1-0.34{)}^5(0.34)=0.04258.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-12. The probability is 0.02009; the expected trial of first success is 2.564. $$P(X=7)=(1-0.39{)}^6(0.39)=0.02009.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-13. The probability is 0.21390; the expected trial of first success is 3.226. $$P(X=2)=(1-0.31{)}^1(0.31)=0.21390.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-14. The probability is 0.13798; the expected trial of first success is 2.273. $$P(X=3)=(1-0.44{)}^2(0.44)=0.13798.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-15. The probability is 0.09925; the expected trial of first success is 3.030. $$P(X=4)=(1-0.33{)}^3(0.33)=0.09925.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-16. The probability is 0.05400; the expected trial of first success is 2.564. $$P(X=5)=(1-0.39{)}^4(0.39)=0.05400.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-17. The probability is 0.05597; the expected trial of first success is 3.704. $$P(X=6)=(1-0.27{)}^5(0.27)=0.05597.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Easy-18. The probability is 0.02640; the expected trial of first success is 2.857. $$P(X=7)=(1-0.35{)}^6(0.35)=0.02640.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-1. The probability is 0.20590; the expected trial of first success is 3.448. $$P(X=2)=(1-0.29{)}^1(0.29)=0.20590.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-2. The probability is 0.14400; the expected trial of first success is 2.500. $$P(X=3)=(1-0.40{)}^2(0.40)=0.14400.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-3. The probability is 0.09612; the expected trial of first success is 2.857. $$P(X=4)=(1-0.35{)}^3(0.35)=0.09612.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-4. The probability is 0.02760; the expected trial of first success is 1.923. $$P(X=5)=(1-0.52{)}^4(0.52)=0.02760.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-5. The probability is 0.02931; the expected trial of first success is 2.439. $$P(X=6)=(1-0.41{)}^5(0.41)=0.02931.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-6. The probability is 0.02009; the expected trial of first success is 2.564. $$P(X=7)=(1-0.39{)}^6(0.39)=0.02009.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-7. The probability is 0.25000; the expected trial of first success is 2.000. $$P(X=2)=(1-0.50{)}^1(0.50)=0.25000.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-8. The probability is 0.13414; the expected trial of first success is 2.174. $$P(X=3)=(1-0.46{)}^2(0.46)=0.13414.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-9. The probability is 0.07727; the expected trial of first success is 2.273. $$P(X=4)=(1-0.44{)}^3(0.44)=0.07727.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-10. The probability is 0.04968; the expected trial of first success is 2.439. $$P(X=5)=(1-0.41{)}^4(0.41)=0.04968.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-11. The probability is 0.04455; the expected trial of first success is 3.030. $$P(X=6)=(1-0.33{)}^5(0.33)=0.04455.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-12. The probability is 0.01141; the expected trial of first success is 2.174. $$P(X=7)=(1-0.46{)}^6(0.46)=0.01141.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-13. The probability is 0.22110; the expected trial of first success is 3.030. $$P(X=2)=(1-0.33{)}^1(0.33)=0.22110.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-14. The probability is 0.14062; the expected trial of first success is 4.000. $$P(X=3)=(1-0.25{)}^2(0.25)=0.14062.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-15. The probability is 0.09612; the expected trial of first success is 2.857. $$P(X=4)=(1-0.35{)}^3(0.35)=0.09612.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-16. The probability is 0.02586; the expected trial of first success is 1.887. $$P(X=5)=(1-0.53{)}^4(0.53)=0.02586.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-17. The probability is 0.01112; the expected trial of first success is 1.852. $$P(X=6)=(1-0.54{)}^5(0.54)=0.01112.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Tough-18. The probability is 0.01599; the expected trial of first success is 2.381. $$P(X=7)=(1-0.42{)}^6(0.42)=0.01599.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-1. The probability is 0.19240; the expected trial of first success is 3.846. $$P(X=2)=(1-0.26{)}^1(0.26)=0.19240.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-2. The probability is 0.12245; the expected trial of first success is 1.961. $$P(X=3)=(1-0.51{)}^2(0.51)=0.12245.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-3. The probability is 0.09252; the expected trial of first success is 2.703. $$P(X=4)=(1-0.37{)}^3(0.37)=0.09252.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-4. The probability is 0.03315; the expected trial of first success is 2.041. $$P(X=5)=(1-0.49{)}^4(0.49)=0.03315.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-5. The probability is 0.01441; the expected trial of first success is 1.961. $$P(X=6)=(1-0.51{)}^5(0.51)=0.01441.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-6. The probability is 0.02474; the expected trial of first success is 2.778. $$P(X=7)=(1-0.36{)}^6(0.36)=0.02474.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-7. The probability is 0.24840; the expected trial of first success is 1.852. $$P(X=2)=(1-0.54{)}^1(0.54)=0.24840.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-8. The probability is 0.14700; the expected trial of first success is 3.333. $$P(X=3)=(1-0.30{)}^2(0.30)=0.14700.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-9. The probability is 0.09437; the expected trial of first success is 2.778. $$P(X=4)=(1-0.36{)}^3(0.36)=0.09437.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-10. The probability is 0.05400; the expected trial of first success is 2.564. $$P(X=5)=(1-0.39{)}^4(0.39)=0.05400.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-11. The probability is 0.01441; the expected trial of first success is 1.961. $$P(X=6)=(1-0.51{)}^5(0.51)=0.01441.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-12. The probability is 0.04449; the expected trial of first success is 4.000. $$P(X=7)=(1-0.25{)}^6(0.25)=0.04449.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-13. The probability is 0.18750; the expected trial of first success is 4.000. $$P(X=2)=(1-0.25{)}^1(0.25)=0.18750.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-14. The probability is 0.14607; the expected trial of first success is 2.632. $$P(X=3)=(1-0.38{)}^2(0.38)=0.14607.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-15. The probability is 0.09252; the expected trial of first success is 2.703. $$P(X=4)=(1-0.37{)}^3(0.37)=0.09252.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-16. The probability is 0.02586; the expected trial of first success is 1.887. $$P(X=5)=(1-0.53{)}^4(0.53)=0.02586.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-17. The probability is 0.03865; the expected trial of first success is 2.778. $$P(X=6)=(1-0.36{)}^5(0.36)=0.03865.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.

Worked solution P46-Toughest-18. The probability is 0.01357; the expected trial of first success is 2.273. $$P(X=7)=(1-0.44{)}^6(0.44)=0.01357.$$ Interpretation: The event requires failures on every earlier trial followed by a success on trial k. Validity: Trials must be independent, use the same p, and continue until the first success. This is legacy enrichment for the revised AP course. Error to reject: Do not use a binomial coefficient because there is only one allowed order for first success on trial k.