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Giải bài 6.5 trang 43 sách bài tập toán 12 - Kết nối tri thức

Cho hai biến cố A và B với \(P\left( A \right) > 0,{\rm{ }}P\left( A \right) > 0\). Chứng minh rằng nếu \(P\left( {AB} \right) = P\left( A \right) \cdot P\left( B \right)\) thì \(A,B\) độc lập.

ꦚTổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Cho hai biến cố A và B với \(P\left( A \right) > 0,{\rm{ }}P\left( A \right) > 0\). Chứng minh rằng nếu \(P\left( {AB} \right) = P\left( A \right) \cdot P\left( B \right)\) thì \(A,B\) độc lập.

Phương pháp giải - Xem chi tiết

Sử dụng công thức tính xác suất có điều kiện để biến đổi, cần chứng minh \(P\left( {A|B} \right) = P\left( A \right)\) và \(P\left( {B|A} \right) = P\left( B \right)\).

Lời giải chi tiết

Giả sử \(P\left( {AB} \right) = P\left( A \right) \cdot P\left( B \right)\) với \(P\left( A \right) > 0,{\rm{ }}P\left( A \right) > 0\). Ta có \(P\left( {A|B} \right) = \frac{{P\left( {AB} \right)}}{{P\left( B \right)}} = \frac{{P\left( A \right) \cdot P\left( B \right)}}{{P\left( B \right)}} = P\left( A \right)\); \(P\left( {B|A} \right) = \frac{{P\left( {AB} \right)}}{{P\left( A \right)}} = \frac{{P\left( A \right) \cdot P\left( B \right)}}{{P\left( A \right)}} = P\left( B \right)\). Suy ra việc xảy ra biến cố B không ảnh hưởng tới xác suất xảy ra của biến cố A và ngược lại. Do đó A và B độc lập.

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