Question

A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.

Answer 1

Let S be the sample space. Then, n(S) = 52.

Let $E_1,E_2$ and $E_3$ be the events of getting a king, a heart and a red card respectivel. Then,

$n\left(E_1\right)=4,n\left(E_2\right)=13$ and $n\left(E_3\right)=26.$

$(E_1\cap E_2)$ = event of getting a king of hearts;

$(E_2\cap E_3)$ = event of getting a heart [$\because$ a heart is a red card also];

$(E_3\cap E_1)$ = event of getting a red king;

and $(E_1\cap E_2\cap E_3)$ = event of getting a king of hearts.

$\therefore n(E_1\cap E_2)=1,n(E_2\cap E_3)=13,n(E_3\cap E_1)=2$ and $n(E_1\cap E_2\cap E_3)=1.$

$\therefore P\left(E_1\right)=\frac{n\left(E_1\right)}{n\left(S\right)}=\frac{4}{52}=\frac{1}{13};P\left(E_2\right)=\frac{n\left(E_2\right)}{n\left(S\right)}=\frac{13}{52}=\frac{1}{4};$

$P\left(E_3\right)=\frac{n\left(E_3\right)}{n\left(S\right)}=\frac{26}{52}=\frac{1}{2};P(E_1\cap E_2)=\frac{n(E_1\cap E_2)}{n\left(S\right)}=\frac{1}{52};$

$P(E_2\cap E_3)=\frac{n(E_2\cap E_3)}{n\left(S\right)}=\frac{2}{52}=\frac{1}{26}$

and $P(E_1\cap E_2\cap E_3)=\frac{n(E_1\cap E_2\cap E_3)}{n\left(S\right)}=\frac{1}{52}.$

$\therefore$ P(getting a king or a heart or a red card)

$\therefore$ P(getting a king or a heart or a red card)

$=P\left(E_{1}or{E}_{2}or{E}_3\right)=P(E_1\cup E_2\cup E_3)$

$=P\left(E_1\right)+P\left(E_2\right)+P\left(E_3\right)-P(E_1\cap E_2)-P(E_2\cap E_3)-P(E_3\cap E_1)+P(E_1\cap E_2\cap E_3)$

$=(\frac{1}{13}+\frac{1}{4}+\frac{1}{2}-\frac{1}{52}-\frac{1}{4}-\frac{1}{26}+\frac{1}{52})=\frac{28}{52}=\frac{7}{13}.$

Hence, the required probability is $\frac{7}{13}.$