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10^{-7}\\right]"},{"type":"text","id":"17","text":"should be 1"},{"type":"expression","id":"7","color":"#2d70b3","latex":"\\frac{1}{N}\\operatorname{total}\\left(F\\right)"},{"type":"text","id":"19","text":"update with success"},{"type":"expression","id":"2","color":"#2d70b3","latex":"U_{S}=\\left(F\\to\\frac{XF}{\\frac{1}{N}\\operatorname{total}\\left(XF\\right)}\\right)"},{"type":"text","id":"22","text":"update with failure"},{"type":"expression","id":"12","color":"#c74440","latex":"Q\\left(x\\right)=1-x","hidden":true},{"type":"expression","id":"13","color":"#2d70b3","latex":"U_{F}=\\left(F\\to\\frac{Q\\left(X\\right)F}{\\frac{1}{N}\\operatorname{total}\\left(Q\\left(X\\right)F\\right)}\\right)"},{"type":"text","id":"46","text":"update with sample"},{"type":"expression","id":"47","color":"#2d70b3","latex":"U_{M}=\\left(F\\to\\frac{X^{n_{s}}Q\\left(X\\right)^{n-n_{s}}F}{\\frac{1}{N}\\operatorname{total}\\left(X^{n_{s}}Q\\left(X\\right)^{n-n_{s}}F\\right)}\\right)"},{"type":"text","id":"24","text":"number of successes and sample size"},{"type":"expression","id":"25","color":"#388c46","latex":"n_{s}=7","slider":{"hardMin":true,"min":"1","step":"1"}},{"type":"expression","id":"11","color":"#000000","latex":"n=10","slider":{"hardMin":true,"hardMax":true,"min":"1","max":"1000","step":"1"}},{"type":"text","id":"51","text":"generate confidence interval"},{"type":"expression","id":"52","color":"#000000","latex":"C=0.9","slider":{"hardMin":true,"hardMax":true,"min":"0","max":"1"}},{"type":"expression","id":"54","color":"#2d70b3","latex":"c_{1}=0.45844665444994426","slider":{"hardMin":true,"hardMax":true,"min":"0","max":"1"}},{"type":"expression","id":"55","color":"#388c46","latex":"c_{2}=0.88252253093988","slider":{"hardMin":true,"hardMax":true,"min":"0","max":"1"}},{"type":"expression","id":"56","color":"#6042a6","latex":"\\left(c_{1},g\\left(c_{1}\\right)\\right)"},{"type":"expression","id":"57","color":"#000000","latex":"\\left(c_{2},g\\left(c_{2}\\right)\\right)"},{"type":"expression","id":"77","color":"#388c46","latex":"\\int_{c_{1}}^{c_{2}}g\\left(x\\right)dx"},{"type":"text","id":"82","text":"bivariate Newton's method magic"},{"type":"expression","id":"76","color":"#2d70b3","latex":"c_{1}\\to c_{1}-\\frac{g\\left(c_{2}\\right)\\left(g\\left(c_{1}\\right)-g\\left(c_{2}\\right)\\right)+g'\\left(c_{2}\\right)\\left(\\int_{c_{1}}^{c_{2}}g\\left(x\\right)dx-C\\right)}{g'\\left(c_{1}\\right)g\\left(c_{2}\\right)-g\\left(c_{1}\\right)g'\\left(c_{2}\\right)},c_{2}\\to c_{2}-\\frac{g\\left(c_{1}\\right)\\left(g\\left(c_{1}\\right)-g\\left(c_{2}\\right)\\right)+g'\\left(c_{1}\\right)\\left(\\int_{c_{1}}^{c_{2}}g\\left(x\\right)dx-C\\right)}{g'\\left(c_{1}\\right)g\\left(c_{2}\\right)-g\\left(c_{1}\\right)g'\\left(c_{2}\\right)}"}]}}