Worked solution P47-Easy-1. The fixed-trial count is binomial with probability 0.11441; the first-success variable is geometric with probability 0.14685. $$\left(\genfrac{}{}{0ex}{}{14}{3}\right)p^3(1-p{)}^{11}=0.11441,(1-p{)}^{2}p=0.14685.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-2. The fixed-trial count is binomial with probability 0.12499; the first-success variable is geometric with probability 0.07243. $$\left(\genfrac{}{}{0ex}{}{13}{4}\right)p^4(1-p{)}^9=0.12499,(1-p{)}^{3}p=0.07243.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-3. The fixed-trial count is binomial with probability 0.21704; the first-success variable is geometric with probability 0.05829. $$\left(\genfrac{}{}{0ex}{}{14}{5}\right)p^5(1-p{)}^9=0.21704,(1-p{)}^{4}p=0.05829.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-4. The fixed-trial count is binomial with probability 0.20095; the first-success variable is geometric with probability 0.02587. $$\left(\genfrac{}{}{0ex}{}{15}{6}\right)p^6(1-p{)}^9=0.20095,(1-p{)}^{5}p=0.02587.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-5. The fixed-trial count is binomial with probability 0.08055; the first-success variable is geometric with probability 0.14607. $$\left(\genfrac{}{}{0ex}{}{15}{3}\right)p^3(1-p{)}^{12}=0.08055,(1-p{)}^{2}p=0.14607.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-6. The fixed-trial count is binomial with probability 0.09630; the first-success variable is geometric with probability 0.07963. $$\left(\genfrac{}{}{0ex}{}{15}{4}\right)p^4(1-p{)}^{11}=0.09630,(1-p{)}^{3}p=0.07963.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-7. The fixed-trial count is binomial with probability 0.21893; the first-success variable is geometric with probability 0.04968. $$\left(\genfrac{}{}{0ex}{}{13}{5}\right)p^5(1-p{)}^8=0.21893,(1-p{)}^{4}p=0.04968.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-8. The fixed-trial count is binomial with probability 0.21109; the first-success variable is geometric with probability 0.02757. $$\left(\genfrac{}{}{0ex}{}{14}{6}\right)p^6(1-p{)}^8=0.21109,(1-p{)}^{5}p=0.02757.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-9. The fixed-trial count is binomial with probability 0.00364; the first-success variable is geometric with probability 0.11426. $$\left(\genfrac{}{}{0ex}{}{16}{3}\right)p^3(1-p{)}^{13}=0.00364,(1-p{)}^{2}p=0.11426.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-10. The fixed-trial count is binomial with probability 0.23669; the first-success variable is geometric with probability 0.09612. $$\left(\genfrac{}{}{0ex}{}{12}{4}\right)p^4(1-p{)}^8=0.23669,(1-p{)}^{3}p=0.09612.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-11. The fixed-trial count is binomial with probability 0.21893; the first-success variable is geometric with probability 0.04968. $$\left(\genfrac{}{}{0ex}{}{13}{5}\right)p^5(1-p{)}^8=0.21893,(1-p{)}^{4}p=0.04968.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-12. The fixed-trial count is binomial with probability 0.16730; the first-success variable is geometric with probability 0.01325. $$\left(\genfrac{}{}{0ex}{}{14}{6}\right)p^6(1-p{)}^8=0.16730,(1-p{)}^{5}p=0.01325.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-13. The fixed-trial count is binomial with probability 0.01040; the first-success variable is geometric with probability 0.12745. $$\left(\genfrac{}{}{0ex}{}{16}{3}\right)p^3(1-p{)}^{13}=0.01040,(1-p{)}^{2}p=0.12745.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-14. The fixed-trial count is binomial with probability 0.04752; the first-success variable is geometric with probability 0.05751. $$\left(\genfrac{}{}{0ex}{}{14}{4}\right)p^4(1-p{)}^{10}=0.04752,(1-p{)}^{3}p=0.05751.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-15. The fixed-trial count is binomial with probability 0.03914; the first-success variable is geometric with probability 0.02418. $$\left(\genfrac{}{}{0ex}{}{16}{5}\right)p^5(1-p{)}^{11}=0.03914,(1-p{)}^{4}p=0.02418.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-16. The fixed-trial count is binomial with probability 0.12415; the first-success variable is geometric with probability 0.01216. $$\left(\genfrac{}{}{0ex}{}{15}{6}\right)p^6(1-p{)}^9=0.12415,(1-p{)}^{5}p=0.01216.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-17. The fixed-trial count is binomial with probability 0.04255; the first-success variable is geometric with probability 0.13971. $$\left(\genfrac{}{}{0ex}{}{15}{3}\right)p^3(1-p{)}^{12}=0.04255,(1-p{)}^{2}p=0.13971.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-18. The fixed-trial count is binomial with probability 0.21284; the first-success variable is geometric with probability 0.08640. $$\left(\genfrac{}{}{0ex}{}{12}{4}\right)p^4(1-p{)}^8=0.21284,(1-p{)}^{3}p=0.08640.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-19. The fixed-trial count is binomial with probability 0.19165; the first-success variable is geometric with probability 0.03911. $$\left(\genfrac{}{}{0ex}{}{13}{5}\right)p^5(1-p{)}^8=0.19165,(1-p{)}^{4}p=0.03911.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-20. The fixed-trial count is binomial with probability 0.21768; the first-success variable is geometric with probability 0.02112. $$\left(\genfrac{}{}{0ex}{}{13}{6}\right)p^6(1-p{)}^7=0.21768,(1-p{)}^{5}p=0.02112.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Easy-21. The fixed-trial count is binomial with probability 0.10075; the first-success variable is geometric with probability 0.14272. $$\left(\genfrac{}{}{0ex}{}{13}{3}\right)p^3(1-p{)}^{10}=0.10075,(1-p{)}^{2}p=0.14272.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-1. The fixed-trial count is binomial with probability 0.04255; the first-success variable is geometric with probability 0.13971. $$\left(\genfrac{}{}{0ex}{}{15}{3}\right)p^3(1-p{)}^{12}=0.04255,(1-p{)}^{2}p=0.13971.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-2. The fixed-trial count is binomial with probability 0.11176; the first-success variable is geometric with probability 0.08852. $$\left(\genfrac{}{}{0ex}{}{16}{4}\right)p^4(1-p{)}^{12}=0.11176,(1-p{)}^{3}p=0.08852.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-3. The fixed-trial count is binomial with probability 0.14757; the first-success variable is geometric with probability 0.02940. $$\left(\genfrac{}{}{0ex}{}{13}{5}\right)p^5(1-p{)}^8=0.14757,(1-p{)}^{4}p=0.02940.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-4. The fixed-trial count is binomial with probability 0.17017; the first-success variable is geometric with probability 0.01825. $$\left(\genfrac{}{}{0ex}{}{15}{6}\right)p^6(1-p{)}^9=0.17017,(1-p{)}^{5}p=0.01825.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-5. The fixed-trial count is binomial with probability 0.14189; the first-success variable is geometric with probability 0.14400. $$\left(\genfrac{}{}{0ex}{}{12}{3}\right)p^3(1-p{)}^9=0.14189,(1-p{)}^{2}p=0.14400.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-6. The fixed-trial count is binomial with probability 0.08542; the first-success variable is geometric with probability 0.06997. $$\left(\genfrac{}{}{0ex}{}{14}{4}\right)p^4(1-p{)}^{10}=0.08542,(1-p{)}^{3}p=0.06997.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-7. The fixed-trial count is binomial with probability 0.19434; the first-success variable is geometric with probability 0.04753. $$\left(\genfrac{}{}{0ex}{}{14}{5}\right)p^5(1-p{)}^9=0.19434,(1-p{)}^{4}p=0.04753.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-8. The fixed-trial count is binomial with probability 0.21238; the first-success variable is geometric with probability 0.02265. $$\left(\genfrac{}{}{0ex}{}{12}{6}\right)p^6(1-p{)}^6=0.21238,(1-p{)}^{5}p=0.02265.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-9. The fixed-trial count is binomial with probability 0.06738; the first-success variable is geometric with probability 0.14129. $$\left(\genfrac{}{}{0ex}{}{14}{3}\right)p^3(1-p{)}^{11}=0.06738,(1-p{)}^{2}p=0.14129.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-10. The fixed-trial count is binomial with probability 0.01389; the first-success variable is geometric with probability 0.05256. $$\left(\genfrac{}{}{0ex}{}{16}{4}\right)p^4(1-p{)}^{12}=0.01389,(1-p{)}^{3}p=0.05256.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-11. The fixed-trial count is binomial with probability 0.15151; the first-success variable is geometric with probability 0.03709. $$\left(\genfrac{}{}{0ex}{}{14}{5}\right)p^5(1-p{)}^9=0.15151,(1-p{)}^{4}p=0.03709.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-12. The fixed-trial count is binomial with probability 0.20101; the first-success variable is geometric with probability 0.03294. $$\left(\genfrac{}{}{0ex}{}{16}{6}\right)p^6(1-p{)}^{10}=0.20101,(1-p{)}^{5}p=0.03294.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-13. The fixed-trial count is binomial with probability 0.01967; the first-success variable is geometric with probability 0.12979. $$\left(\genfrac{}{}{0ex}{}{15}{3}\right)p^3(1-p{)}^{12}=0.01967,(1-p{)}^{2}p=0.12979.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-14. The fixed-trial count is binomial with probability 0.17503; the first-success variable is geometric with probability 0.08421. $$\left(\genfrac{}{}{0ex}{}{13}{4}\right)p^4(1-p{)}^9=0.17503,(1-p{)}^{3}p=0.08421.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-15. The fixed-trial count is binomial with probability 0.10296; the first-success variable is geometric with probability 0.02760. $$\left(\genfrac{}{}{0ex}{}{14}{5}\right)p^5(1-p{)}^9=0.10296,(1-p{)}^{4}p=0.02760.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-16. The fixed-trial count is binomial with probability 0.21768; the first-success variable is geometric with probability 0.02112. $$\left(\genfrac{}{}{0ex}{}{13}{6}\right)p^6(1-p{)}^7=0.21768,(1-p{)}^{5}p=0.02112.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-17. The fixed-trial count is binomial with probability 0.01967; the first-success variable is geometric with probability 0.12979. $$\left(\genfrac{}{}{0ex}{}{15}{3}\right)p^3(1-p{)}^{12}=0.01967,(1-p{)}^{2}p=0.12979.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-18. The fixed-trial count is binomial with probability 0.05447; the first-success variable is geometric with probability 0.06749. $$\left(\genfrac{}{}{0ex}{}{15}{4}\right)p^4(1-p{)}^{11}=0.05447,(1-p{)}^{3}p=0.06749.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-19. The fixed-trial count is binomial with probability 0.19971; the first-success variable is geometric with probability 0.05615. $$\left(\genfrac{}{}{0ex}{}{15}{5}\right)p^5(1-p{)}^{10}=0.19971,(1-p{)}^{4}p=0.05615.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-20. The fixed-trial count is binomial with probability 0.16428; the first-success variable is geometric with probability 0.03865. $$\left(\genfrac{}{}{0ex}{}{13}{6}\right)p^6(1-p{)}^7=0.16428,(1-p{)}^{5}p=0.03865.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Tough-21. The fixed-trial count is binomial with probability 0.19537; the first-success variable is geometric with probability 0.14788. $$\left(\genfrac{}{}{0ex}{}{12}{3}\right)p^3(1-p{)}^9=0.19537,(1-p{)}^{2}p=0.14788.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-1. The fixed-trial count is binomial with probability 0.04255; the first-success variable is geometric with probability 0.13971. $$\left(\genfrac{}{}{0ex}{}{15}{3}\right)p^3(1-p{)}^{12}=0.04255,(1-p{)}^{2}p=0.13971.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-2. The fixed-trial count is binomial with probability 0.07877; the first-success variable is geometric with probability 0.06000. $$\left(\genfrac{}{}{0ex}{}{13}{4}\right)p^4(1-p{)}^9=0.07877,(1-p{)}^{3}p=0.06000.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-3. The fixed-trial count is binomial with probability 0.15151; the first-success variable is geometric with probability 0.03709. $$\left(\genfrac{}{}{0ex}{}{14}{5}\right)p^5(1-p{)}^9=0.15151,(1-p{)}^{4}p=0.03709.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-4. The fixed-trial count is binomial with probability 0.17658; the first-success variable is geometric with probability 0.03110. $$\left(\genfrac{}{}{0ex}{}{12}{6}\right)p^6(1-p{)}^6=0.17658,(1-p{)}^{5}p=0.03110.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-5. The fixed-trial count is binomial with probability 0.19537; the first-success variable is geometric with probability 0.14788. $$\left(\genfrac{}{}{0ex}{}{12}{3}\right)p^3(1-p{)}^9=0.19537,(1-p{)}^{2}p=0.14788.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-6. The fixed-trial count is binomial with probability 0.21634; the first-success variable is geometric with probability 0.09437. $$\left(\genfrac{}{}{0ex}{}{13}{4}\right)p^4(1-p{)}^9=0.21634,(1-p{)}^{3}p=0.09437.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-7. The fixed-trial count is binomial with probability 0.19971; the first-success variable is geometric with probability 0.05615. $$\left(\genfrac{}{}{0ex}{}{15}{5}\right)p^5(1-p{)}^{10}=0.19971,(1-p{)}^{4}p=0.05615.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-8. The fixed-trial count is binomial with probability 0.20101; the first-success variable is geometric with probability 0.03294. $$\left(\genfrac{}{}{0ex}{}{16}{6}\right)p^6(1-p{)}^{10}=0.20101,(1-p{)}^{5}p=0.03294.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-9. The fixed-trial count is binomial with probability 0.15384; the first-success variable is geometric with probability 0.14746. $$\left(\genfrac{}{}{0ex}{}{13}{3}\right)p^3(1-p{)}^{10}=0.15384,(1-p{)}^{2}p=0.14746.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-10. The fixed-trial count is binomial with probability 0.02663; the first-success variable is geometric with probability 0.05503. $$\left(\genfrac{}{}{0ex}{}{15}{4}\right)p^4(1-p{)}^{11}=0.02663,(1-p{)}^{3}p=0.05503.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-11. The fixed-trial count is binomial with probability 0.21904; the first-success variable is geometric with probability 0.06040. $$\left(\genfrac{}{}{0ex}{}{13}{5}\right)p^5(1-p{)}^8=0.21904,(1-p{)}^{4}p=0.06040.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-12. The fixed-trial count is binomial with probability 0.20285; the first-success variable is geometric with probability 0.02931. $$\left(\genfrac{}{}{0ex}{}{13}{6}\right)p^6(1-p{)}^7=0.20285,(1-p{)}^{5}p=0.02931.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-13. The fixed-trial count is binomial with probability 0.14189; the first-success variable is geometric with probability 0.14400. $$\left(\genfrac{}{}{0ex}{}{12}{3}\right)p^3(1-p{)}^9=0.14189,(1-p{)}^{2}p=0.14400.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-14. The fixed-trial count is binomial with probability 0.08542; the first-success variable is geometric with probability 0.06997. $$\left(\genfrac{}{}{0ex}{}{14}{4}\right)p^4(1-p{)}^{10}=0.08542,(1-p{)}^{3}p=0.06997.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-15. The fixed-trial count is binomial with probability 0.22250; the first-success variable is geometric with probability 0.04118. $$\left(\genfrac{}{}{0ex}{}{12}{5}\right)p^5(1-p{)}^7=0.22250,(1-p{)}^{4}p=0.04118.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-16. The fixed-trial count is binomial with probability 0.16428; the first-success variable is geometric with probability 0.03865. $$\left(\genfrac{}{}{0ex}{}{13}{6}\right)p^6(1-p{)}^7=0.16428,(1-p{)}^{5}p=0.03865.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-17. The fixed-trial count is binomial with probability 0.14189; the first-success variable is geometric with probability 0.14400. $$\left(\genfrac{}{}{0ex}{}{12}{3}\right)p^3(1-p{)}^9=0.14189,(1-p{)}^{2}p=0.14400.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-18. The fixed-trial count is binomial with probability 0.06488; the first-success variable is geometric with probability 0.07727. $$\left(\genfrac{}{}{0ex}{}{16}{4}\right)p^4(1-p{)}^{12}=0.06488,(1-p{)}^{3}p=0.07727.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-19. The fixed-trial count is binomial with probability 0.07491; the first-success variable is geometric with probability 0.03315. $$\left(\genfrac{}{}{0ex}{}{16}{5}\right)p^5(1-p{)}^{11}=0.07491,(1-p{)}^{4}p=0.03315.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-20. The fixed-trial count is binomial with probability 0.17658; the first-success variable is geometric with probability 0.03110. $$\left(\genfrac{}{}{0ex}{}{12}{6}\right)p^6(1-p{)}^6=0.17658,(1-p{)}^{5}p=0.03110.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Worked solution P47-Toughest-21. The fixed-trial count is binomial with probability 0.19537; the first-success variable is geometric with probability 0.14788. $$\left(\genfrac{}{}{0ex}{}{12}{3}\right)p^3(1-p{)}^9=0.19537,(1-p{)}^{2}p=0.14788.$$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.