问答题
对飞机进行三次独立射击，第一次射击命中率为0.4，第二次为0.5，第三次为0.7.击中飞机一次而飞机被击落的概率为0.2，击中飞机两次而飞机被击落的概率为0.6，若被击中三次，则飞机必被击落．求射击三次飞机未被击落的概率．
【正确答案】
【答案解析】令A
i
=“恰有i次击中飞机”，i=0，1，2，3，B=“飞机被击落”．
P(A 0 )=(1-0.4)(1-0.5)(1-0.7)=0.09
P(A 1 )=0.4×(1-0.5)×(1-0.7)+(1-0.4)×0.5×(1-0.7)+(1-0.4)×(1-0.5)×0.7
=0.36
P(A 2 )=0.4×0.5×(1-0.7)+0.4×(1-0.5)×0.7+(1-0.4)×0.5×0.7
=0.41
P(A 3 )=0.4×0.5×0.7=0.14
而P(B|A 0 )=0，P(B|A 1 )=0.2，P(B|A 2 )=0.6，P(B|A 3 )=1
所以
P(A 0 )=(1-0.4)(1-0.5)(1-0.7)=0.09
P(A 1 )=0.4×(1-0.5)×(1-0.7)+(1-0.4)×0.5×(1-0.7)+(1-0.4)×(1-0.5)×0.7
=0.36
P(A 2 )=0.4×0.5×(1-0.7)+0.4×(1-0.5)×0.7+(1-0.4)×0.5×0.7
=0.41
P(A 3 )=0.4×0.5×0.7=0.14
而P(B|A 0 )=0，P(B|A 1 )=0.2，P(B|A 2 )=0.6，P(B|A 3 )=1
所以