Question

Two cards are drawn at random from a standard 52-card deck without replacement. Find the probability that the second card drawn is a spade, given that the first card drawn was not a spade.

Answer 1

Probability that second card drawn is a spade given that the first card drawn was not a spade =

= P(second card drawn is a spade | first card drawn was not a spade)

= P( second card drawn is a spade and first card drawn was not a spade ) / P(first card drawn was not a spade)

Now,

P(first card drawn was not a spade) = Number of non spade cards / Total cards

Number of spade cards = 13

Number of non-spade cards = 52 - 13 = 39

P(first card drawn was not a spade) = 39 / 52 = 3 / 4

Now,

Since first card has been drawn and it was a non-spade card

Number of spade cards remaining = 13

Number of total cards remaining = 51

P( second card drawn is a spade and first card drawn was not a spade ) =

= Number of spade cards remaining / Total cards remaining

= 13 / 51

So,

P(second card drawn is a spade | first card drawn was not a spade) = ( 13 / 51 ) ( 3 / 4 )

= 13 / 68 = 0.1912

Hence,

Probability that second card drawn is a spade given that the first card drawn was not a spade = 0.1912