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Giải bài 4 trang 100 sách bài tập toán 11 - Chân trời sáng tạo tập 2

Cho A và B là hai biến cố độc lập với nhau. a) Biết \(P\left( A \right) = 0,4\) và \(P\left( {\overline A B} \right) = 0,3\). b) Biết \(P\left( {\overline A B} \right) = 0,4\) và \(P\left( {A \cup B} \right) = 0,9\).

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Đề bài

Cho A và B là hai biến cố độc lập với nhau. a) Biết \(P\left( A \right) = 0,4\) và \(P\left( {\overline A B} \right) = 0,3\). Tính xác suất của các biến cố B và \(A \cup B\). b) Biết \(P\left( {\overline A B} \right) = 0,4\) và \(P\left( {A \cup B} \right) = 0,9\). Tính xác suất của các biến cố A, B, AB.

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về quy tắc cộng cho hai biến cố bất kì: Cho hai biến cố A và B. Khi đó, \(P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {AB} \right)\). 

Lời giải chi tiết

a) \(P\left( A \right) \) \( = 0,4 \) \( \Rightarrow P\left( {\overline A } \right) \) \( = 0,6\) Vì A và B là hai biến cố độc lập với nhau nên \(P\left( {\overline A B} \right) \) \( = P\left( {\overline A } \right)P\left( B \right) \) \( = 0,3 \) \( \Rightarrow P\left( B \right) \) \( = \frac{{0,3}}{{0,6}} \) \( = 0,5\), \(P\left( {AB} \right) \) \( = P\left( A \right)P\left( B \right) \) \( = 0,4.0,5 \) \( = 0,2\) Do đó, \(P\left( {A \cup B} \right) \) \( = P\left( A \right) + P\left( B \right) - P\left( {AB} \right) \) \( = 0,4 + 0,5 - 0,2 \) \( = 0,7\) b) Vì A và B là hai biến cố độc lập với nhau nên \(P\left( {\overline A B} \right) \) \( = P\left( {\overline A } \right)P\left( B \right) \) \( = 0,4 \) \( \Rightarrow \left[ {1 - P\left( A \right)} \right]P\left( B \right) \) \( = 0,4 \) \( \Rightarrow P\left( B \right) - P\left( A \right)P\left( B \right) \) \( = 0,4\) Lại có: \(P\left( {A \cup B} \right) \) \( = P\left( A \right) + P\left( B \right) - P\left( {AB} \right) \) \( = 0,9 \) \( \Rightarrow P\left( A \right) + 0,4 \) \( = 0,9 \) \( \Rightarrow P\left( A \right) \) \( = 0,5\) Suy ra: \(P\left( {\overline A } \right) \) \( = 0,5 \) \( \Rightarrow P\left( B \right) \) \( = \frac{{0,4}}{{P\left( {\overline A } \right)}} \) \( = 0,8\), \(P\left( {AB} \right) \) \( = P\left( A \right).P\left( B \right) \) \( = 0,8.0,5 \) \( = 0,4\)

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