{"status": "success", "data": {"description_md": "There are $6$ peas in a glass, $4$ floating on the top and $2$ sitting on the bottom. At each five second interval, a random number of peas from $0$ to $2$ sink from the top to the bottom and a random number from $0$ to $2$ rise from the bottom to the top. (If there is only $1$ pea left, it moves or stays put with equal probability.) If the probability that all six peas are on the top before all six are on the bottom can be expressed as $\\frac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.", "description_html": "<p>There are <span class=\"katex--inline\">6</span> peas in a glass, <span class=\"katex--inline\">4</span> floating on the top and <span class=\"katex--inline\">2</span> sitting on the bottom. At each five second interval, a random number of peas from <span class=\"katex--inline\">0</span> to <span class=\"katex--inline\">2</span> sink from the top to the bottom and a random number from <span class=\"katex--inline\">0</span> to <span class=\"katex--inline\">2</span> rise from the bottom to the top. (If there is only <span class=\"katex--inline\">1</span> pea left, it moves or stays put with equal probability.) If the probability that all six peas are on the top before all six are on the bottom can be expressed as <span class=\"katex--inline\">\\frac{m}{n}</span>, where <span class=\"katex--inline\">m,n</span> are relatively prime positive integers, find <span class=\"katex--inline\">m+n</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Problem of the Day #65", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}