Question

If $P\left(A\right)=\frac{1}{8}$ and $P\left(B\right)=\frac{5}{8}.$ Which of the following statements is not correct?

• A. $P(A\cup B)\le \frac{3}{4}$
• B. $P(A\cap B)\le \frac{1}{8}$
• C. $P(\overline{A}\cap B)\le \frac{5}{8}$
• D. None of the above

Answer 1

Answer: (A), (B), (C)

The correct options are
A $P(A\cup B)\le \frac{3}{4}$
B $P(\overline{A}\cap B)\le \frac{5}{8}$
C $P(A\cap B)\le \frac{1}{8}$

Since, $P(A\cup B)\le P\left(A\right)+P\left(B\right)$
$\Rightarrow P(A\cup B)\le \frac{1}{8}+\frac{5}{8}$
$\Rightarrow P(A\cup B)\le \frac{3}{4}$
Hence (A) is correct
Again $P(A\cap B)\le P\left(A\right)$
$\Rightarrow P(A\cap B)\le \frac{1}{8}....\left(1\right)$
also $P(A\cap B)\le P\left(B\right)$
$\Rightarrow P(A\cap B)\le \frac{5}{8}.....\left(2\right)$
Hence (B) is also correct
Now $P(A\cap B)\le 0$
$P\left(B\right)-P(\overline{A}\cap B)\ge 0$
$\Rightarrow P(\overline{A}\cap B)\le P\left(B\right)$
$\Rightarrow P(\overline{A}\cup B)\le \frac{5}{8}$
Hence (C) is also correct.