A single card is drawn from a pack of $52$ cards. What is the probability that
- the card is a jack or a king;
- the card will be $5$ or smaller.
$1$ card from a pack of $52$ cards can be drawn in $52$ ways.
$\therefore \;$ Number of elements in sample space $S = n \left(S\right) = 52$
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Let $A$ be the event that the selected card is a jack or a king
$1$ jack can be selected from $4$ jacks in ${^{4}}{C}_{1} = 4$ ways
$\hspace{3cm}$ OR
$1$ king can be selected from $4$ kings in ${^{4}}{C}_{1} = 4$ ways
$\therefore \;$ Number of elements in $A = n\left(A\right) = 4 + 4 = 8$
$\therefore \;$ Probability of event $A = P\left(A\right) = \dfrac{n\left(A\right)}{n \left(S\right)} = \dfrac{8}{52} = \dfrac{2}{13}$
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Let $B$ be the event that the selected card is $5$ or smaller
There are $16$ such cards $\;\;\;$ (2, 3, 4, 5 of each suite)
$1$ card from $16$ cards can be selected in $16$ ways
$\therefore \;$ Number of elements in $B = n\left(B\right) = 16$
$\therefore \;$ Probability of event $B = P\left(B\right) = \dfrac{n\left(B\right)}{n \left(S\right)} = \dfrac{16}{52} = \dfrac{4}{13}$