4. 托马斯·贝叶斯(
ThomasBayes)在研究“逆向概率”的问题中得到了一个公式:
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmrow%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%7C%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%7C%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmi%3EA%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%7C%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmi%3EA%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%7C%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmsup%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmsup%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,这个公式被称为贝叶斯公式(贝叶斯定理),其中
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%7C%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmi%3EA%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%7C%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmsup%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%E2%8B%85%3C%2Fmo%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmsup%3E%3Cmi%3EA%3C%2Fmi%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
称为
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EB%3C%2Fmi%3E%3C%2Fmath%3E)
的全概率.这个定理在实际生活中有着重要的应用价值.假设某种疾病在所有人群中的感染率是0.1%,医院现有的技术对于该疾病检测准确率为99%,即已知患病情况下,99%的可能性可以检查出阳性,正常人
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmn%3E99%3C%2Fmn%3E%3Cmi%3E%25%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的可能性检查为正常.如果从人群中随机抽一个人去检测,经计算检测结果为阳性的全概率为0.01098,请你用贝叶斯公式估计在医院给出的检测结果为阳性的条件下这个人得病的概率( )