# A card is drawn from a standard deck. A second card is drawn, without replacing the first card. What is the probability that the first card is red and the second card is black?

##### 1 Answer

In the standard deck of cards there are 52 cards, 26 **red** and 26 **black**.

SOLUTION #1

The first event, *random drawing* of a **red** card (event *sample space* of an entire deck with 52 different *elementary events* occurring with equal probabilities of **red**). Therefore, the probability of picking a **red** card equals to

The second event, *random drawing* of a **black** card from a deck of only 51 remaining cards (event **red** card is drawn), the deck for the second drawing contains 25 **red** and 26 **black** cards and the *conditional probability* of picking **black** card is

Now we can use the concept of *conditional probability* and a formula that describes the probability of a combined event through the probability of one and conditional probability of another:

In our case

Therefore,

SOLUTION #2

After shuffling there are

Let's count only those where there is a **red** card on the first place and **black** card on the second.

There are 26 **red** and 26 **black** cards. Therefore, we have

Therefore, we have **red** and the second is **black**.

The final probability, therefore, is

Informative lectures and solutions to many problems of the Theory of Probabilities for beginners can be found in the corresponding chapter on the Web site Unizor (free online course of advanced mathematics for teenagers).