Example 10
Find the probability that when a hand of 7 cards is drawn from a well
shuffled deck of 52 cards, it contains
(i) all Kings
7 cards are to be chosen from 52 cards
Total number of combinations (hands) possible = 52C7
= 52!/7!(52 −7)! = 52!/7!45!
Let A be the event that all kings are selected
There are only 4 kings in a pack of 52 cards
Hence if 7 cards are chosen
4 kings to be chosen out of 4 and 3 others to be chosen out of remaining 48
Hence total number of combinations
n(A) = 4C4 × 48C3
= 4!/4!0! × 48!/3!(48 −3)! = 1× 48/(3! 45!) = 48!/(3! 45!)
Hence
P (A) = (𝑛(𝐴))/(𝑛(𝑆))
= 48!/(3! 45!) ÷ 52!/(7! 45!)
= (48! × 7!)/(3! × 52!) = 𝟏/𝟕𝟕𝟑𝟓
Example 10
Find the probability that when a hand of 7 cards is drawn from a well
shuffled deck of 52 cards, it contains
(ii) 3 Kings
Let B be that event that 3 king are selected
There are only 4 king in a pack of 52 cards
Hence if 7 cards are chosen,
3 king to be chosen out of 4 and 4 other to be chosen out of remaining 48
Hence, number of combination
n(B) = 4C3 × 48C4
= 4!/3!(4 −3)! × 48!/4!(48 −4)!
= 4 × 48!/4!44!
Hence,
P(B) = (𝑛(𝐵))/(𝑛(𝑆))
= (4 × 48!/4!44!)/(52!/7!45!)
= (4 × 48!)/4!44! × 7!45!/52!
= (4 × 48! × 7! × 45!)/(4! × 44! × 52!)
= (4 × 48! × 7! × 45!)/(4! × 44! × 52 × 51 × 50 × 49 × 48!)
= (4 × 7! × 45)/(4! × 52 × 51 × 50 × 49 )
= (4 × 7 × 6 × 5 × 4! × 45)/( 52 × 51 × 50 × 49 × 4!)
= 𝟗/𝟏𝟓𝟒𝟕
Example 10
Find the probability that when a hand of 7 cards is drawn from a well
shuffled deck of 52 cards, it contains
(iii) at least 3 Kings.
Atleast 3 kings are selected means
either 3 kings are selected or 4 kings are selected
So, P(at least 3 king) = P(3 King) + ( 4 King)
We know that,
P(3 Kings) = 9/1547
P(4 Kings) = 1/7735
(calculated in (ii) part)
(calculated in (i) part)
P(at least 3 king) = 9/1547 + 1/7735
= 𝟒𝟔/𝟕𝟕𝟑𝟓

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!