Two cards are drawn together (simultaneously) at random from a well-shuffled pack of 52 playing cards. What is the probability that one card is a spade and the other is a heart?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\frac{13}{102}\)
Drawing two cards together is a selection, so outcomes are counted with combinations. One spade and one heart means choosing 1 of the 13 spades and 1 of the 13 hearts, so the two counts are multiplied.
Ways to draw 2 cards \(\displaystyle ={}^{52}C_2=\frac{52\times51}{2}=1326\)
Ways to draw 1 spade and 1 heart \(={}^{13}C_1\times{}^{13}C_1=13\times13=169\)
Probability \(\displaystyle =\frac{169}{1326}=\frac{13}{102}\) (dividing both by 13)