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Find the probability that when a hand of 7 cards are drawn from the well-shuffled deck of 52 cards, it contains (i) all kings (ii) 3 kings Find the probability that when a hand of 7 cards are drawn from the well-shuffled deck of 52 cards, it contains(i) all kings (ii) 3 kings
Find the probability that when a hand of 7 cards are drawn from the well-shuffled deck of 52 cards, it contains
(i) all kings (ii) 3 kings
Dear Student(i) To find the probability that all the cards are kings:If 7 cards are chosen from the pack of 52 cardsThen the total number of combinations possible are:52C7= 52!/[7! (52-7)!]= 52!/ (7! 45!)Assume that A be the event that all the kings are selectedWe know that there are only 4 kings in the pack of 52 cardsThus, if 7 cards are chosen, 4 kings are chosen out of 4, and 3 should be chosen form the 48 remaining cards.Therefore, the total number of combinations is:n(A) =4C4x48C3= [4!/4!0! ] x [48!/3!(48-3)!]= 1 x [48!/3! 45!]= 48!/3! 45!Therefore, P(A) = n(A)/n(S)= [48!/3! 45!] ÷[52!/ (7! 45!]= [48! x 7!] ÷ [3!x 52!]= 1/7735Therefore, the probability of getting all the 7 cards are kings is 1/7735(ii) To find the probability that 3 cards are kings:Assume that B be the event that 3 kings are selected.Thus, if 7 cards are chosen, 3 kings are chosen out of 4, and 4 cards should be chosen form the 48 remaining cards.Therefore, the total number of combinations is:n(B) =4C3x48C4= [4!/3!(4-3)! ] x [48!/4!(48-4)!]= 4 x [48!/4! 44!]= 48!/3! 45!Therefore, P(B) = n(B)/n(S)= [4 x48!/4! 44!] ÷[52!/ (7! 45!]= 9/1547Therefore, the probability of getting 3 kings is 9/1547Thanks
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