One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent ?
$\mathrm{E}:$ ' the card drawn is black '
$\mathrm{F}:$ ' the card drawn is a king '
One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent ?
$\mathrm{E}:$ ' the card drawn is black '
$\mathrm{F}:$ ' the card drawn is a king '
In a deck of $52$ cards, $26$ cards are black and $4$ cards are kings.
$\therefore $ $\mathrm{P}(\mathrm{E})=\mathrm{P}$ (the card drawn is a black ) $=\frac{26}{52}=\frac{1}{2}$
$\therefore $ $\mathrm{P}(\mathrm{F})=\mathrm{P}$ (the card drawn is a king ) $=\frac{4}{52}=\frac{1}{13}$
In the pack of $52$ cards, $2$ cards are black as well as kings.
$\therefore $ $\mathrm{P}(\mathrm{EF})=\mathrm{P}$ (the card drawn is black king ) $=\frac{2}{52}=\frac{1}{26}$
$\mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{1}{2} \cdot \frac{1}{13}=\frac{1}{26}=\mathrm{P}(\mathrm{EF})$
Therefore, the given events $\mathrm{E}$ and $\mathrm{F}$ are independent.
Similar Questions
Events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12}$ and $\mathrm{P}$ $($ not $ \mathrm{A}$ or not $\mathrm{B})=\frac{1}{4} .$ State whether $\mathrm{A}$ and $\mathrm{B}$ are independent?
Events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12}$ and $\mathrm{P}$ $($ not $ \mathrm{A}$ or not $\mathrm{B})=\frac{1}{4} .$ State whether $\mathrm{A}$ and $\mathrm{B}$ are independent?
If $P(A) = 0.25,\,\,P(B) = 0.50$ and $P(A \cap B) = 0.14,$ then $P(A \cap \bar B)$ is equal to
If $P(A) = 0.25,\,\,P(B) = 0.50$ and $P(A \cap B) = 0.14,$ then $P(A \cap \bar B)$ is equal to
The probabilities that $A$ and $B$ will die within a year are $p$ and $q$ respectively, then the probability that only one of them will be alive at the end of the year is
The probabilities that $A$ and $B$ will die within a year are $p$ and $q$ respectively, then the probability that only one of them will be alive at the end of the year is
The probabilities that a student passes in Mathematics, Physics and Chemistry are $m, p$ and $c$ respectively. On these subjects, the student has a $75\%$ chance of passing in at least one, a $50\%$ chance of passing in at least two and a $40\%$ chance of passing in exactly two. Which of the following relations are true
The probabilities that a student passes in Mathematics, Physics and Chemistry are $m, p$ and $c$ respectively. On these subjects, the student has a $75\%$ chance of passing in at least one, a $50\%$ chance of passing in at least two and a $40\%$ chance of passing in exactly two. Which of the following relations are true
- [IIT 1999]
If from each of the three boxes containing $3$ white and $1$ black, $2$ white and $2$ black, $1$ white and $3$ black balls, one ball is drawn at random, then the probability that $2$ white and $1$ black ball will be drawn is
If from each of the three boxes containing $3$ white and $1$ black, $2$ white and $2$ black, $1$ white and $3$ black balls, one ball is drawn at random, then the probability that $2$ white and $1$ black ball will be drawn is
- [IIT 1998]