Answer with Step-by-step explanation:
We are given that
P(A)=0.4 and P(B)=0.7
We know that
[tex]P(A)+P(B)+P(A\cap B)=P(A\cup B)[/tex]
We know that
Maximum value of [tex]P(A\cup B)[/tex]=1 and minimum value of [tex]P(A\cup B)[/tex]=0
[tex]0\leq P(A\cup B )\leq 1[/tex]
[tex]0\leq P(A)+P(B)-P(A\cap B)\leq 1[/tex]
[tex]0\leq 0.4+0.7-P(A\cap B)\leq 1[/tex]
[tex]0\leq 1.1-P(A\cap B)\leq 1[/tex]
[tex]0\leq 1.1-P(A\cap B)[/tex]
[tex]P(A\cap B)\leq 1.1[/tex]
It is not possible that [tex]P(A\cap B)[/tex] is equal to 1.1
[tex]1.1-P(A\cap B)\leq 1[/tex]
[tex]-P(A\cap B)\leq 1-1.1=-0.1[/tex]
Multiply by (-1) on both sides
[tex]P(A\cap B)\geq 0.1[/tex]
Again, [tex]P(A\cup B)\geq P(B)[/tex]
[tex]0.4+0.7-P(A\cap B)\geq 0.7[/tex]
[tex]1.1-P(A\cap B)\geq 0.7[/tex]
[tex]-P(A\cap B)\geq -1.1+0.7=-0.4[/tex]
Multiply by (-1) on both sides
[tex]P(A\cap B)\leq 0.4[/tex]
Hence, [tex]0.1\leq P(A\cap B)\leq 0.4[/tex]
