3 cards are drawn from the deck of play cards and I want to define the correct complementary event for the event where all drawn cards are suited.

Question

3 cards are drawn from the deck of the 52 cards with club, spade, heart, diamond(13 cards respectively).

$$\begin{align} E_1&:=\text{event where that 3 drawn cards have same suits} \\P_1&:=\text{probability of }~E_1\\ \end{align}$$

The text book says that the complementary event for $~ E_1 ~$ is the event where 3 pairwise distinct suits with that 3 drawn cards.

And the book shows the following eqn of the probability of the complementary event for the answer.

$$ 1\times {39 \over 51 }\times {26 \over 50 } = {1014 \over 2550 } $$

In the first place I think this definition of the complementary event is wrong. I think that the correct complementary event for $~ E_1 ~$ is as following.

$$ E_{2}:=\text{event where number of types of suits is greater than or equal to}~2 $$

I've stopped proceeding calculations for $~ P(E_2) ~$ because I can't hold a confidence that the book made the mistake.

Who is correct actually?

Answer 1

The book's thought process can be inferred from the equation:

1. When drawing the first card, there are 52 cards, of which 52 are not of a suit seen before.
2. When drawing the second card, there are 51 remaining cards, of which 39 are not of a suit seen before.
3. When drawing the third card, there are 50 remaining cards, of which 26 are not of a suit seen before.

Thus the book is describing the chance of all three cards being of unique suits. As noted, this is not the complementary event, since it does not account for the possibility of drawing two cards from one suit and one card from another.