{"status": "success", "data": {"description_md": "**POTD November 21, 2023**\n**Author:** munch\n\nHeather and Thomas play a game involving counters, in which a coin is flipped repeatedly. If the coin lands heads, then Thomas gives Heather a counter, and if the coin lands tails, then Heather gives Thomas a counter. Both Heather and Thomas start with two counters each, and the winner is the first person who acquires all four counters. If the coin is weighted so that the probability of heads is $\\frac{3}{5}$, and the probability of tails is $\\frac{2}{5}$, then the probability that Heather wins the game can be expressed in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.", "description_html": "<p><strong>POTD November 21, 2023</strong><br/>&#10;<strong>Author:</strong> munch</p>&#10;<p>Heather and Thomas play a game involving counters, in which a coin is flipped repeatedly. If the coin lands heads, then Thomas gives Heather a counter, and if the coin lands tails, then Heather gives Thomas a counter. Both Heather and Thomas start with two counters each, and the winner is the first person who acquires all four counters. If the coin is weighted so that the probability of heads is <span class=\"katex--inline\">\\frac{3}{5}</span>, and the probability of tails is <span class=\"katex--inline\">\\frac{2}{5}</span>, then the probability that Heather wins the game can be expressed in the form <span class=\"katex--inline\">\\frac{m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers. Find <span class=\"katex--inline\">m+n</span>.</p>&#10;", "hints_md": "Let $P(k)$ be the probability that if Heather starts with $k$ coins, she wins. Then $P(0) = 0$ and $P(4) = 1 ...$", "hints_html": "<p>Let <span class=\"katex--inline\">P(k)</span> be the probability that if Heather starts with <span class=\"katex--inline\">k</span> coins, she wins. Then <span class=\"katex--inline\">P(0) = 0</span> and <span class=\"katex--inline\">P(4) = 1 ...</span></p>&#10;", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Problem of the Day #2", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}