Question 1201400
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Problem 1
P(both kings or both black cards) 


There are 4 kings
There are 52 cards total
4/52 = 1/13 = probability of getting a king
Assuming the card is not put back (aka no replacement), then
4-1 = 3 kings left
52-1 = 51 cards left
3/51 = 1/17 = probability of getting another king, no replacement
P(2 kings) = P(1st king)*P(2nd king)
P(2 kings) = (4/52)*(3/51)
P(2 kings) = (1/13)*(1/17)
P(2 kings) = 1/221


26 black cards (13 spades + 13 clubs)
52 cards total
26/52 = 1/2 = probability of getting a black card
26-1 = 25 black cards left
52-1 = 51 cards left
25/51 = probability of getting another black card, no replacement
P(2 black cards) = P(1st black card)*P(2nd black card)
P(2 black cards) = (26/52)*(25/51)
P(2 black cards) = (1/2)*(25/51)
P(2 black cards) = 25/102


2 black kings
52 cards total
P(2 black kings) = P(1st black king)*P(2nd black king)
P(2 black kings) = (2/52)*(1/51)
P(2 black kings) = 1/1326


P(2 kings or 2 black) = P(2 kings) + P(2 black) - P(2 black kings)
P(2 kings or 2 black) = 1/221  +  25/102  -  1/1326
P(2 kings or 2 black) = 6/1326 + 325/1326 - 1/1326
P(2 kings or 2 black) = (6 + 325 - 1)/1326
P(2 kings or 2 black) = 330/1326
P(2 kings or 2 black) = (6*55)/(6*221)
P(2 kings or 2 black) = <font color=red>55/221</font>


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Problem 2
P(both face cards or both red)


face cards = Jack, Queen, King
3 face cards in any given suit
4*3 = 12 face cards in the deck


P(2 face cards) = P(1st face card)P*(2nd face card)
P(2 face cards) = (12/52)*(11/51)
P(2 face cards) = (3/13)*(11/51)
P(2 face cards) = 11/221
I'm assuming that we are not replacing the 1st card.


26 red cards (13 hearts + 13 diamonds)
P(2 red cards) = P(1st red card)*P(2nd red card)
P(2 red cards) = (26/52)*(25/51)
P(2 red cards) = (1/2)*(25/51)
P(2 red cards) = 25/102


6 cards are red and a face card (eg: King of Diamonds)
P(2 red face cards) = P(1st red face)*P(2nd red face)
P(2 red face cards) = (6/52)*(5/51)
P(2 red face cards) = (3/26)*(5/51)
P(2 red face cards) = 15/1326
P(2 red face cards) = 5/442


P(2 face or 2 red) = P(2 face) + P(2 red) - P(2 red face)
P(2 face or 2 red) = 11/221  +  25/102  -  5/442
P(2 face or 2 red) = 66/1326 + 325/1326 - 15/1326
P(2 face or 2 red) = (66 + 325 - 15)/1326
P(2 face or 2 red) = 376/1326
P(2 face or 2 red) = (2*188)/(2*663)
P(2 face or 2 red) = <font color=red>188/663</font>


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Summary


P(both kings or both black cards) = <font color=red>55/221</font>
P(both face cards or both red) = <font color=red>188/663</font>


The 1st card wasn't put back, aka no replacement.
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