**Toolbox:**

- If \( p(A)p(B)=p(A\cap B)\)
- Then Aand Bare independent
- die is thrown 2 times
- sample space={(1,1)(1,2).....(6,6)} 36 such events
- A=same number in each throw
- and (1,1)(2,2)(3,3)(4,4) (5,5)(6,6)
- B=total number is 10 or more
- = (5,5)(6,4)(4,6)(6,5)(5,6)(6,6)
- \( A\cap B\) ={(5,5)(6,6)}

\( A\cap B\) ={(5,5)(6,6)}\)

\(=(0.2\times0.2)+(0.2\times0.2)+(0.1)\times(0.1) +(0.4)(0.4)+(0.1)(0.1)+0.1\times0.1\)

\( =0.04+0.04+0.01+0.16+0.01+0.01\)

\(P(A)=0.27\)

\(p(B)=p(5,5)+p(6,4)+.....+p(6,6)\)

\(p(B)=(0.1\times0.1)=0.1\times0.3+0.3\times0.1+0.1\times0.1+0.1\times0.1\)

\(=0.01+0.03+0.03+0.03+0.01+0.01\)

\(=0.09\)

\(p(A\cap\; B)=p(5,5)+p(6,6)\)

\(=0.1\times0.1+0.1\times0.1\)

\(=0.01=0.01\)

\(=0.02\)

\(p(A)\;\times\;p(B)=0.27\times0.09=0.0213\)

\(p(A)p(B)\neq p(A\cap B)\)

so A and B are not indipendent