Question

If $P\left(A\right)=0.65,P\left(B\right)=0.80$ then $P\left(A\cap B\right)$ lies in the interval:

• A. $\left[0.30,0.80\right]$
• B. $\left[0.35,0.75\right]$
• C. $\left[0.4,0.70\right]$
• D. $\left[0.45,0.65\right]$

Answer 1

Answer: (D)

$\left[0.45,0.65\right]$

We have,

$P\left(A\right)=0.65$

$P\left(B\right)=0.80$

We know that

$P(A\cap B)\le min(P\left(A\right),P\left(B\right))$

So,

$P(A\cap B)\le min(0.65,0.80)$

$P(A\cap B)\le 0.65$

We know that,

$P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$

But $P(A\cup B)\le 1$

Therefore,

$P(A\cap B)\ge 0.65+0.80-1$

$P(A\cap B)\ge 0.45$

Therefore,

$0.45\le P(A\cap B)\le 0.65$

Hence, $P(A\cap B)$ lies in $[0.45,0.65]$.