Question

You choose a card from a deck of 52 cards. Without replacing the card, you choose a second card. What is the probability of choosing two red cards?

Answer 1

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

The first event, random drawing of a red card (event A), has a sample space of an entire deck with 52 different elementary events occurring with equal probabilities of

1/52 . Out of them only 26 events are "good" (that is, the card we randomly pick is red). Therefore, the probability of picking a red card equals to P(A) = 26/52 = 1/2

The second event, random drawing of a black card from a deck of only 51 remaining cards (event B), is dependent on the results of the first event. If the first event (event A) occurs (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 P(B ∣ A) = 26/51.

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: P(A ⋅ B ) = P( A) ⋅ P(B ∣ A)

In our case P(A) = 1/2 and P(B ∣ A) = 26/51


Therefore, P(A ⋅ B) = (1/2) ⋅ (26/51) = 13/51

Answer 2

Answer: 13/51

Step-by-step explanation: After shuffling there are 52 permutations of 52 cards. Each has the probability of occurrence equal to another, that is the probability of any sequence of cards is 1/52.

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 26 ⋅26 choices for a pair of two first cards. The sequence of other 50 cards in unimportant and any permutation among them is fine for us since we fixed the first two.

Therefore, we have 26 ⋅ 26 ⋅50 different permutations when the first card is red and the second is black.

Therefore, we have 26 ⋅ 26 ⋅50 different permutations when the first card is red and the second is black.

The final probability, therefore, is

26 ⋅26 ⋅50 / 52 = 26 ⋅26 / 52 ⋅ 51 = 13 / 51

13/51 is the final answer