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Q.

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

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Detailed Solution

(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

P( non f ace )= Number of favourable outcomes  Total Number of outcomes =4052=1013P( non face )=1013

Therefore, the probability of getting “non-face cards” is 1013.

 

(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

P( black king )= Number of favourable outcomes  Total Number of outcomes =252=126P( black king )=126

Therefore, the probability of getting “a black king” is 126.

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