{"status": "success", "data": {"description_md": "In a small pond there are eleven lily pads in a row labeled 0 through 10.  A frog is sitting on pad 1.  When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\\frac{N}{10}$ and to pad $N+1$ with probability $1-\\frac{N}{10}$.  Each jump is independent of the previous jumps.  If the frog reaches pad 0 it will be eaten by a patiently waiting snake.  If the frog reaches pad 10 it will exit the pond, never to return.  What is the probability that the frog will escape without being eaten by the snake?\n\n$\\textbf{(A) }\\frac{32}{79}\\qquad<br>\\textbf{(B) }\\frac{161}{384}\\qquad<br>\\textbf{(C) }\\frac{63}{146}\\qquad<br>\\textbf{(D) }\\frac{7}{16}\\qquad<br>\\textbf{(E) }\\frac{1}{2}\\qquad$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>In a small pond there are eleven lily pads in a row labeled 0 through 10.  A frog is sitting on pad 1.  When the frog is on pad  <span class=\"katex--inline\">N</span> ,  <span class=\"katex--inline\">0&lt;N&lt;10</span> , it will jump to pad  <span class=\"katex--inline\">N-1</span>  with probability  <span class=\"katex--inline\">\\frac{N}{10}</span>  and to pad  <span class=\"katex--inline\">N+1</span>  with probability  <span class=\"katex--inline\">1-\\frac{N}{10}</span> .  Each jump is independent of the previous jumps.  If the frog reaches pad 0 it will be eaten by a patiently waiting snake.  If the frog reaches pad 10 it will exit the pond, never to return.  What is the probability that the frog will escape without being eaten by the snake?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }\\frac{32}{79}\\qquad\\textbf{(B) }\\frac{161}{384}\\qquad\\textbf{(C) }\\frac{63}{146}\\qquad\\textbf{(D) }\\frac{7}{16}\\qquad\\textbf{(E) }\\frac{1}{2}\\qquad</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2014 AMC 12B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/14_amc12B_p23", "prev": "/problem/14_amc12B_p21"}}