Fill in the blanks.

Question 11 Mark

A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number on the die and a spade card is ____________ .

Answer

$\frac{1}{8}$, because
Let $E_1=$ Getting an even number on die
$E_2=A$ card selected is a spade card
$\therefore P\left(E_1\right)=\frac{3}{6}=\frac{1}{2}$ and $P\left(E_2\right)=\frac{13}{52}=\frac{1}{4}$
Then, $\quad P\left(E_1 \cap E_2\right)=P\left(E_1\right) \cdot P\left(E_2\right)$
$=\frac{1}{2} \cdot \frac{1}{4}=\frac{1}{8}$
[Since, $E_1$ and $E_2$ and independent events]

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Question 21 Mark

Let $P(A)=\frac{7}{13}, P(B)=\frac{9}{13}$ and $P(A \cap B)=\frac{4}{13}$. Then $P\left(\frac{A^{\prime}}{B}\right)$ is equal to ____________ .

Answer

$\frac{5}{9}$, because
$P\left(\frac{A^{\prime}}{B}\right)=\frac{P\left(A^{\prime} \cap B\right)}{P(B)}$
$=\frac{P(B)-P(A \cap B)}{P(B)}$
$=\frac{\frac{9}{13}-\frac{4}{13}}{\frac{9}{13}}=\frac{5}{9}$

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Question 31 Mark

Let $A$ and $B$ be two given mutually exclusive events. Then $P\left(\frac{A}{B}\right)$ is ____________ .

Answer

0, because
$\begin{array}{r}P(A \cap B)=0 (\because A \text { and } B \text { are mutually exclusive events})\end{array}$
$\therefore \quad P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}=\frac{0}{P(B)}=0$.

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Question 41 Mark

If $A$ and $B$ are two events associated with a random experiment and $P\left(\frac{A}{B}\right)=P(A)$, then $A$ is ____________ of $B$.

Answer

independent, because
Given, $\quad P\left(\frac{A}{B}\right)=P(A)$
$\Rightarrow \quad \frac{P(A \cap B)}{P(B)}=P(A)$
$\Rightarrow \quad P(A \cap B)=P(A) \cdot P(B)$
So, A and B are independent events.

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