One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting : 

(i) a non-face card, (ii) a black king

(𝑖) 𝑎 𝑛𝑜𝑛 − 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑 

We need to find the probability of getting a non-face card from a deck of 52 cards. 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 − 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 = 52 𝑇𝑜𝑡𝑎𝑙 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 𝑖𝑛 𝑑𝑒𝑐𝑘 = 12 

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛 − 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 = 52 − 12 

⇒ 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛 − 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 = 40

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 52 

𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛 − 𝑓𝑎𝑐𝑒 𝑐𝑎𝑟𝑑𝑠 

𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 40 

Therefore, the probability of getting ‘non face cards’ is

$\begin{array}{r}\Rightarrow P\left(\text{ non }f\text{ ace }\right)=\frac{\text{ Number of favourable outcomes }}{\text{ Total Number of outcomes }}\\ =\frac{40}{52}\\ =\frac{10}{13}\\ \Rightarrow P\left(\text{ non face }\right)=\frac{10}{13}\end{array}$

Therefore, the probability of getting “non-face cards” is $\frac{10}{13}$.

(𝑖𝑖) 𝑎 𝑏𝑙𝑎𝑐𝑘 𝑘𝑖𝑛𝑔 

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 52

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑙𝑎𝑐𝑘 𝑘𝑖𝑛𝑔𝑠 = 2 

⇒ 𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 2

Therefore, the probability of getting ‘a black king’ is

$\begin{array}{r}P\left(\text{ black king }\right)=\frac{\text{ Number of favourable outcomes }}{\text{ Total Number of outcomes }}\\ =\frac{2}{52}\\ =\frac{1}{26}\\ \Rightarrow P\left(\text{ black king }\right)=\frac{1}{26}\end{array}$

Therefore, the probability of getting “a black king” is $\frac{1}{26}$.