Question 1201400: Two cards are drawn from a standard deck of cards. Find each probability
P(both kings or both black cards)
P(both face cards or both red)
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
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) = 55/221
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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) = 188/663
=====================================================
Summary
P(both kings or both black cards) = 55/221
P(both face cards or both red) = 188/663
The 1st card wasn't put back, aka no replacement.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
I personally find it easier to count the numbers of ways of getting the desired results and then calculating the probability, rather than working with probabilities to do all the calculations.
In counting the number of ways of getting the desired results, we use the inclusion-exclusion principle. For example, in the first problem where the desired outcome is 2 kings or 2 face cards, we count the number of ways of getting 2 kings, add the number of ways of getting 2 black cards, and then subtract the number of ways we counted twice -- the ways that get 2 cards that are BOTH black AND kings.
Then the other problem is worked in the same way.
For both problems: Number of ways of choosing 2 of the 52 cards: C(52,2) = 1326
(1) P(both kings or both black)
# of ways of getting 2 black cards: C(26,2) = 325
# of ways of getting 2 kings: C(4,2) = 6
# of ways of getting 2 black kings: C(2,2) = 1
# of ways of getting 2 black cards or 2 kings: 325+6-1 = 330
ANSWER: P(both kings or both black) = 330/1326 = 55/221
(2) P(both face cards or both red)
# of ways of getting 2 red cards: C(26,2) = 325
# of ways of getting 2 face cards: C(12,2) = 66
# of ways of getting 2 face cards both red: C(6,2) = 15
# of ways of getting both face cards or both red: 325+66-15 = 376
ANSWER: P(both face cards or both kings) = 376/1326 = 188/663
Same answers as from the other tutor, by a different path.
Neither path is better than the other; try both and find the one that "works" best for you.
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