Suppose that four cards are drawn as specified below from an

Suppose that four cards are drawn as specified below from an ordinary deck of 52 randomly shuffled cards. Find the probability that: (a) At least one King appears (assume \"with replacement\");
(b) At least one King appears (assume \"without replacement\");
(c) The cards are all of one suit and at most 1 is a face card (assume \"with replacement\");
(d) The cards are all of one suit and at most 1 is a face card (assume \"without replacement\");

Solution

(a) At least one King appears [ WITH REPLACEMENT]

Drawing the 4 cards from pack of 52 = 52^4
Drawing the cards with No kings picked = 48^4
P(Drawing the cards atleast with 1 King ) = 1 - p( pick cards with no king) = 1 - 48^4 / 52^4 = 0.2739

(b) At least one King appears

Drawing the 4 cards from pack of 52 = 52 C 4
Drawing the cards with No kings picked = 48 C 4
P(Drawing the cards atleast with 1 King ) = 1 - p( pick cards with no king) = 1 - 48 C 4 / 52 C 4 = 0.2812

(c) [ WITH REPLACEMENT]
Drawing the card with same suit = We have 4 diffrent suits, picking any one suit gives the probability to 1/4. selcetion of a card 4 times is (13)^4, the chances are = 1/ 4 * (13)^4
Drawing the card with same suit atmost 1 is face card= P ( Drawing 4 cards with zero face card ) + P( Drawing 4 cards with atmost 1 face card) = (1/ 4 * (9)^4)/ (52)^4 + (1/ 4 * (9)^3 * (4)^1) / (52)^4 = 0.000324

(d) [ WITHOUT REPLACEMENT]
Drawing the card with same suit = We have 4 diffrent suits, picking any one suit gives the probability to 1/4. selcetion of a card 4 times is 13 C 4, the chances are = 1/ 4 * 13 C 4
Drawing the card with same suit atmost 1 is face card = P ( Drawing 4 cards with zero face card ) + P( Drawing 4 cards with atmost 1 face card) = (1/ 4 * 9 C 4 )/ 52 C 4+ (1/ 4 * 9 C 3 * 4 C 1 ) / 52 C 4 = 0.0004266