$I=\int\limits_{0}^{\frac{\pi}{4} }\frac{\sin x+\cos x}{3+\sin 2x} dx =\frac{1}{4}\int\limits_{0}^{\frac{\pi}{4} }(\sin x+\cos x)\frac{2-\sin x+\cos x+2+\sin x-\cos x}{4-(\sin x- \cos x)^2} dx$
$=\frac{1}{4}\int\limits_{0}^{\frac{\pi}{4} }\frac{\sin x+\cos x}{\sin x-\cos x+2} dx+\frac{1}{4}\int\limits_{0}^{\frac{\pi}{4} }\frac{\sin x+\cos x}{-\sin x+\cos x+2} dx$
 $=\frac{1}{4}\ln\left| {\sin x-\cos x+2} \right|_{0}^{\frac{\pi}{4} }-\frac{1}{4}\ln\left| {-\sin x+\cos x+2} \right|_{0}^{\frac{\pi}{4} }$
 $I=\frac{1}{4}\ln3$

3.
a) $\Omega=\left\{ {(i,j)|1 \le i,j \le 4} \right\}\Rightarrow |\Omega|=16$
b)
$A=\left\{ {(1,1),(1,3),(2,2),(2,4),(3,1),(3,3),(4,1),(4,3)} \right\}\Rightarrow P(A)=\frac{8}{16}$
$B=\left\{ {(1,2),(1,4),(2,1),(2,2),(2,3),(2,4),(3,2),(3,4),(4,1),(4,2),(4,3),(4,4)} \right\}\Rightarrow P(B)=\frac{12}{16}$

6.
$\Omega=\left\{ {1,2,3,4,5,6} \right\}, |\Omega|=6$
a) $A=\left\{ {2,4,6} \right\}, P(A)=\frac{3}{6}$
a) $B=\left\{ {3,6} \right\}, P(B)=\frac{2}{6}$
a) $C=\left\{ {3,4,5,6} \right\}, P(C)=\frac{4}{6}$

7.
$\Omega=\left\{ {SS,SN,NS,NN} \right\}, |\Omega|=4$
a) $A=\left\{ {SS} \right\}, P(A)=\frac{1}{4}$
a) $B=\left\{ {SN,NS} \right\}, P(B)=\frac{2}{4}$
a) $C=\left\{ {SS,SN,NS} \right\}, P(C)=\frac{3}{4}$

4.
a)
$A = \overline{A_1}\cap\overline{A_2}, C= ({A_1}\cap\overline{A_2})\cup( \overline{A_1}\cap{A_2})$
$B={A_1}\cap{A_2}, D={A_1}\cup{A_2}$
b) $\overline{D}$ là biến cố : Cả hai người đều bắn trượt.
Như vậy $\overline{D} =\overline{A_1}\cap\overline{A_2}=A$
 $B \cap C=\emptyset$ nên $B$ và $C$ xung khắc.

2.
a) $\Omega=\left\{ {(I,J)|1 \le i,j \le 6} \right\}$
b)
A : Lần gieo đầu xuất hiện mặt $6$
B: Tổng số chấm trong hai lần gieo là $8$
C: Kết quả của hai lần gieo như nhau

1.
a) $\Omega=\left\{ {SSS,SSN,NSS,SNS,NNS,NSN,SNN,NNN} \right\}$
b)
$A=\left\{ {SSS,SSN,SNS,SNN} \right\}$
$B=\left\{ {SNN,NSN,NNS} \right\}$
$C=\left\{ {SSS,NSS,SNS,NNS,NSN,SNN,NNN} \right\}$

5.
a) $\Omega=\left\{ {1,2,3,\cdots,50} \right\}\Rightarrow |\Omega|=50$
b) $A=\left\{ {2,3,5,\cdots,47} \right\}\Rightarrow |A|=15$
 c) $P(A)=\frac{15}{50}$
 d) $B=\left\{ {1,2,3} \right\}\Rightarrow |B|=3$
 $P(B)=\frac{3}{50}$