logs/step_110-b_0.log

{
  "log_file": [
    "step_110-b_0.log",
    "step_110-b_1.log",
    "step_110-b_2.log",
    "step_110-b_3.log",
    "step_110-b_4.log",
    "step_110-b_5.log"
  ],
  "query": [
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.\n\n### Response:\nSEARCH[\"\\\\frac{1}{10}.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/25051.json\n\n#### Similar Question\nRural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\\frac{1}{10}$ of the boys and $\\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent?\n\n#### User Answer\n$\\frac{1}{10}$ of $60$ boys is $60/10=6$ students, while $\\frac{1}{3}$ of $45$ girls is $45/3=15$ students, so $21$ students were absent that day. Since we know that $\\frac{21}{105}=\\frac{1}{5}$ and $\\frac{1}{5}$ is equal to $20\\%$, we know that $20 \\%$ of the total student population was absent.\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.\n\n### Response:\nSEARCH[\"\\\\frac{1}{10}.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/2031.json\n\n#### Similar Question\nFind the fraction that equals $0.\\overline{4}$.\n\n#### User Answer\nWe have \\[0.\\overline{4} = \\frac{4}{10} + \\frac{4}{100} + \\frac{4}{1000} + \\cdots.\\]This infinite geometric series has first term $4/10=2/5$ and common ratio $1/10$, so we have \\[0.\\overline{4} = \\frac{2/5}{1-1/10} = \\frac{4}{9}.\\]\n\n--- RESULTS END ---\n\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.\n\n### Response:\n",
    "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.\n\n### Response:\nSEARCH[\"\\\\frac{1}{10}.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/precalculus/496.json\n\n#### Similar Question\nWe define two sequences of vectors $(\\mathbf{v}_n)$ and $(\\mathbf{w}_n)$ as follows: First, $\\mathbf{v}_0 = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix},$ $\\mathbf{w}_0 = \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}.$  Then for all $n \\ge 1,$ $\\mathbf{v}_n$ is the projection of $\\mathbf{w}_{n - 1}$ onto $\\mathbf{v}_0,$ and $\\mathbf{w}_n$ is the projection of $\\mathbf{v}_n$ onto $\\mathbf{w}_0.$  Find\n\\[\\mathbf{v}_1 + \\mathbf{v}_2 + \\mathbf{v}_3 + \\dotsb.\\]\n\n#### User Answer\nSince $\\mathbf{v}_n$ is always a projection onto $\\mathbf{v}_0,$\n\\[\\mathbf{v}_n = a_n \\mathbf{v}_0\\]for some constant $a_n.$  Similarly,\n\\[\\mathbf{w}_n = b_n \\mathbf{w}_0\\]for some constant $b_n.$\n\n[asy]\nunitsize(1.5 cm);\n\npair[] V, W;\n\nV[0] = (1,3);\nW[0] = (4,0);\nV[1] = (W[0] + reflect((0,0),V[0])*(W[0]))/2;\nW[1] = (V[1] + reflect((0,0),W[0])*(V[1]))/2;\nV[2] = (W[1] + reflect((0,0),V[0])*(W[1]))/2;\nW[2] = (V[2] + reflect((0,0),W[0])*(V[2]))/2;\nV[3] = (W[2] + reflect((0,0),V[0])*(W[2]))/2;\nW[3] = (V[3] + reflect((0,0),W[0])*(V[3]))/2;\n\ndraw((-1,0)--(5,0));\ndraw((0,-1)--(0,4));\ndraw((0,0)--V[0],red,Arrow(6));\ndraw((0,0)--W[0],red,Arrow(6));\ndraw((0,0)--V[1],red,Arrow(6));\ndraw((0,0)--W[1],red,Arrow(6));\ndraw((0,0)--V[2],red,Arrow(6));\ndraw((0,0)--W[2],red,Arrow(6));\ndraw(W[0]--V[1]--W[1]--V[2]--W[2],dashed);\n\nlabel(\"$\\mathbf{v}_0$\", V[0], NE);\nlabel(\"$\\mathbf{v}_1$\", V[1], NW);\nlabel(\"$\\mathbf{v}_2$\", V[2], NW);\nlabel(\"$\\mathbf{w}_0$\", W[0], S);\nlabel(\"$\\mathbf{w}_1$\", W[1], S);\nlabel(\"$\\mathbf{w}_2$\", W[2], S);\n[/asy]\n\nThen\n\\begin{align*}\n\\mathbf{v}_n &= \\operatorname{proj}_{\\mathbf{v}_0} \\mathbf{w}_{n - 1} \\\\\n&= \\frac{\\mathbf{w}_{n - 1} \\cdot \\mathbf{v}_0}{\\|\\mathbf{v}_0\\|^2} \\mathbf{v}_0 \\\\\n&= \\frac{b_{n - 1} \\cdot \\mathbf{w}_0 \\cdot \\mathbf{v}_0}{\\|\\mathbf{v}_0\\|^2} \\mathbf{v}_0 \\\\\n&= \\frac{b_{n - 1} \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix}}{\\left\\| \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix} \\right\\|^2} \\mathbf{v}_0 \\\\\n&= \\frac{2}{5} b_{n - 1} \\mathbf{v}_0.\n\\end{align*}Thus, $a_n = \\frac{2}{5} b_{n - 1}.$\n\nSimilarly,\n\\begin{align*...\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"\\\\frac{1}{10}.\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe probability that it rains on exactly 0, 1, or 2 days is $\\binom{30}{0}\\left(\\frac{1}{10}\\right)^0\\left(\\frac{9}{10}\\right)^30+\\binom{30}{1}\\left(\\frac{1}{10}\\right)^1\\left(\\frac{9}{10}\\right)^29+\\binom{30}{2}\\left(\\frac{1}{10}\\right)^2\\left(\\frac{9}{10}\\right)^29=\\boxed{0.3119}$.</s>",
    " SEARCH[\"\\\\frac{1}{10}.\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe probability that it rains on exactly 1 day is $\\binom{30}{1}\\left(\\frac{1}{10}\\right)=\\frac{30}{10}$ and the probability that it rains on exactly 2 days is $\\binom{30}{2}\\left(\\frac{1}{10}\\right)=\\frac{45}{2}$. Thus, the probability that it rains on at most 2 days is $\\frac{30}{10}+\\frac{45}{2}=\\boxed{11.5}$.</s>",
    " SEARCH[\"\\\\frac{1}{10}.\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe probability that it rains on exactly one day is $\\binom{30}{1}\\left(\\frac{1}{10}\\right)^1\\left(\\frac{9}{10}\\right)^{30-1}=90$.  The probability that it rains on exactly two days is $\\binom{30}{2}\\left(\\frac{1}{10}\\right)^2\\left(\\frac{9}{10}\\right)^{30-2}=360$.  So the probability that it rains on at most two days is $90+360=\\boxed{450}$.</s>"
  ],
  "src_path": [
    "train/counting_and_probability/187.json",
    "train/counting_and_probability/187.json",
    "train/counting_and_probability/187.json",
    "train/counting_and_probability/187.json",
    "train/counting_and_probability/187.json",
    "train/counting_and_probability/187.json"
  ],
  "input": [
    "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.",
    "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.",
    "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.",
    "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.",
    "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth.",
    "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$.  What is the probability that it rains on at most 2 days in June?  Express your answer as a decimal to the nearest thousandth."
  ],
  "output": [
    "There are 30 days in June. The probability that it rains on exactly 0, 1, or 2 days is  \\begin{align*}&\\ \\ \\ \\ \\binom{30}{0}\\bigg(\\frac{1}{10}\\bigg)^{\\!0}\\bigg(\\frac{9}{10}\\bigg)^{\\!30}\\\\&+\\binom{30}{1}\\bigg(\\frac{1}{10}\\bigg)^{\\!1}\\bigg(\\frac{9}{10}\\bigg)^{\\!29}\\\\&+\\binom{30}{2}\\bigg(\\frac{1}{10}\\bigg)^{\\!2}\\bigg(\\frac{9}{10}\\bigg)^{\\!28} \\\\\n&\\approx \\boxed{0.411}.\\end{align*}",
    "There are 30 days in June. The probability that it rains on exactly 0, 1, or 2 days is  \\begin{align*}&\\ \\ \\ \\ \\binom{30}{0}\\bigg(\\frac{1}{10}\\bigg)^{\\!0}\\bigg(\\frac{9}{10}\\bigg)^{\\!30}\\\\&+\\binom{30}{1}\\bigg(\\frac{1}{10}\\bigg)^{\\!1}\\bigg(\\frac{9}{10}\\bigg)^{\\!29}\\\\&+\\binom{30}{2}\\bigg(\\frac{1}{10}\\bigg)^{\\!2}\\bigg(\\frac{9}{10}\\bigg)^{\\!28} \\\\\n&\\approx \\boxed{0.411}.\\end{align*}",
    "There are 30 days in June. The probability that it rains on exactly 0, 1, or 2 days is  \\begin{align*}&\\ \\ \\ \\ \\binom{30}{0}\\bigg(\\frac{1}{10}\\bigg)^{\\!0}\\bigg(\\frac{9}{10}\\bigg)^{\\!30}\\\\&+\\binom{30}{1}\\bigg(\\frac{1}{10}\\bigg)^{\\!1}\\bigg(\\frac{9}{10}\\bigg)^{\\!29}\\\\&+\\binom{30}{2}\\bigg(\\frac{1}{10}\\bigg)^{\\!2}\\bigg(\\frac{9}{10}\\bigg)^{\\!28} \\\\\n&\\approx \\boxed{0.411}.\\end{align*}",
    "There are 30 days in June. The probability that it rains on exactly 0, 1, or 2 days is  \\begin{align*}&\\ \\ \\ \\ \\binom{30}{0}\\bigg(\\frac{1}{10}\\bigg)^{\\!0}\\bigg(\\frac{9}{10}\\bigg)^{\\!30}\\\\&+\\binom{30}{1}\\bigg(\\frac{1}{10}\\bigg)^{\\!1}\\bigg(\\frac{9}{10}\\bigg)^{\\!29}\\\\&+\\binom{30}{2}\\bigg(\\frac{1}{10}\\bigg)^{\\!2}\\bigg(\\frac{9}{10}\\bigg)^{\\!28} \\\\\n&\\approx \\boxed{0.411}.\\end{align*}",
    "There are 30 days in June. The probability that it rains on exactly 0, 1, or 2 days is  \\begin{align*}&\\ \\ \\ \\ \\binom{30}{0}\\bigg(\\frac{1}{10}\\bigg)^{\\!0}\\bigg(\\frac{9}{10}\\bigg)^{\\!30}\\\\&+\\binom{30}{1}\\bigg(\\frac{1}{10}\\bigg)^{\\!1}\\bigg(\\frac{9}{10}\\bigg)^{\\!29}\\\\&+\\binom{30}{2}\\bigg(\\frac{1}{10}\\bigg)^{\\!2}\\bigg(\\frac{9}{10}\\bigg)^{\\!28} \\\\\n&\\approx \\boxed{0.411}.\\end{align*}",
    "There are 30 days in June. The probability that it rains on exactly 0, 1, or 2 days is  \\begin{align*}&\\ \\ \\ \\ \\binom{30}{0}\\bigg(\\frac{1}{10}\\bigg)^{\\!0}\\bigg(\\frac{9}{10}\\bigg)^{\\!30}\\\\&+\\binom{30}{1}\\bigg(\\frac{1}{10}\\bigg)^{\\!1}\\bigg(\\frac{9}{10}\\bigg)^{\\!29}\\\\&+\\binom{30}{2}\\bigg(\\frac{1}{10}\\bigg)^{\\!2}\\bigg(\\frac{9}{10}\\bigg)^{\\!28} \\\\\n&\\approx \\boxed{0.411}.\\end{align*}"
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