Two cards are drawn from a deck of 52 cards without replacement. what is the probability of getting both cards black or getting one card queen and the other king?
Answers
let b be the event that we get both black cards
n( b)= 26
thus p(b)= 26÷52=1 upon 2
let a be the even that we get one queen and one King
thus n(a) = 16
p(a)=16÷52= 8/26 = 4/13
Given: Two cards are drawn from a deck of 52 cards without replacement.
To find: Probability of getting both cards black or getting one card queen and the other king
Solution: A deck of cards has 52 cards. Out of these 52 cards, 26 are black and 26 are red. There are 12 face cards- 4 queens, 4 kings and 4 jacks in a deck of 52 cards.
Probability of getting both cards black
Since there are 26 black cards, probability of getting black card in first draw
= Number of black cards/ Total cards
= 26/52
= 1/2
= 0.5
Now, the card is not replaced with another card. Therefore, number of cards= 51 and number of black cards= 25
Probability of getting black card in second draw
= Number of black cards/ Total cards
= 25/51
Probability of getting both cards black
= Probability of getting black in first draw× Probability of getting black card in draw
= 0.5 × 25/51
= 25/102
Probability of getting one card queen and the other card king
There are two cases here:
a- first drawn card is king and second drawn card is queen
Probability of getting a king in first draw
= Number of kings/ Total cards
= 4/52
= 1/13
Since the card is not replaced, total cards now= 51
Probability of getting queen in second draw
= Number of queens / Total cards
= 4/51
Probability of getting a king in first draw and queen in second draw
= 1/13 × 4/51
= 4/663
b- first drawn card is queen and second drawn card is king
Probability of getting a queen in first draw
= Number of queens/ Total cards
= 4/52
= 1/13
Since the card is not replaced, total cards now= 51
Probability of getting king in second draw
= Number of kings / Total cards
= 4/51
Probability of getting a queen in first draw and king in second draw
= 1/13 × 4/51
= 4/663
Therefore, probability of getting a queen and a king
= Probability of getting a king in first draw and queen in second draw + Probability of getting a queen in first draw and king in second draw
= 4/663 + 4/663
= 8/663
Therefore, probability of getting both cards black or getting one card queen and the other king
= probability of getting both cards black + probability of getting one card queen and the other king
= 25/ 102 + 8/663
= 0.245 + 0.012
= 0.257
Therefore, probability of getting both cards black or getting one card queen and the other king is 0.257.