题目

已知某高射炮对敌机进行射击，其击中敌机发动机、机舱及其他部位的概率分别为0.1,0.08,0.39。又若击中上述部位而使飞机坠落的概率分别为0.1,0.08,0.39。现在该炮任意发射一炮弹，问飞机坠落的概率为（）(保留小数点后两位，四舍五入)0.1,0.08,0.390.1,0.08,0.390.1,0.08,0.390.1,0.08,0.39

已知某高射炮对敌机进行射击，其击中敌机发动机、机舱及其他部位的概率分别为 $0.1,0.08,0.39$ 。又若击中上述部位而使飞机坠落的概率分别为 $0.95,0.89,0.51$ 。现在该炮任意发射一炮弹，问飞机坠落的概率为（）(保留小数点后两位，四舍五入)

$A.0.36$

$B.0.37$

$C.0.43$

$D.0.39$

题目解答

解：

记 $A_1,A_2,A_3$ 分别为击中敌机发动机、机舱及其他部位，事件 $B$ 为被击中后飞机坠落，则由题可知 $P\left(A_1\right)=0.1,P\left(A_2\right)=0.08,P\left(A_3\right)=0.39$

$P\left(B|A_1\right)=0.95,P\left(B|A_2\right)=0.89,P\left(B|A_3\right)=0.51$

由全概率公式可得

$P\left(B\right)=P\left(A_1B\right)+P\left(A_2B\right)+P\left(A_3B\right)$

由贝叶斯公式可得

$P\left(A_1B\right)=P\left(A_1\right)\cdot P\left(B|A_1\right)$

$P\left(A_2B\right)=P\left(A_2\right)\cdot P\left(B|A_2\right)$

$P\left(A_3B\right)=P\left(A_3\right)\cdot P\left(B|A_3\right)$

∴飞机坠落的概率为： $P\left(B\right)=P\left(A_1B\right)+P\left(A_2B\right)+P\left(A_3B\right)$

$=P\left(A_1\right)\cdot P\left(B|A_1\right)+P\left(A_2\right)\cdot P\left(B|A_2\right)$

$+P\left(A_3\right)\cdot P\left(B|A_3\right)$

$=0.1\times0.95+0.08\times0.89+0.39\times0.51$

$=0.3651\approx0.37$

选择 $B$