Two cards are drawn at random from a standard 52-card deck of playing cards without replacement. Find the probability that the first card is a heart and the second is red.

13/52 + 25/51 = ?

13/52 * 25/51 = 25/204

To find the probability of drawing a heart as the first card and a red card as the second card, we need to determine the number of favorable outcomes and the total number of possible outcomes.

1. Determining the number of favorable outcomes:
- There are 13 hearts in a standard deck of cards.
- After drawing a heart as the first card, there will be 51 cards remaining in the deck, with 26 red cards (13 hearts + 13 diamonds).
- Hence, the number of favorable outcomes is 13 (for the first card being a heart) multiplied by 26 (for the second card being red).

Number of favorable outcomes: 13 * 26 = 338

2. Determining the total number of possible outcomes:
- After drawing one card, there are 51 cards remaining in the deck.
- Hence, the total number of possible outcomes is 52 (for the first card) multiplied by 51 (for the second card).

Total number of possible outcomes: 52 * 51 = 2652

3. Calculating the probability:
- The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 338 / 2652

Therefore, the probability that the first card is a heart and the second card is red is 338 / 2652, which can be further simplified if needed.

To find the probability that the first card is a heart and the second is red, we need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break down the problem step by step:

Step 1: Determine the number of favorable outcomes.
The first card being a heart means that we need to choose one card from the 13 hearts in the deck. Therefore, the number of favorable outcomes for the first card is 13.

After the first card is drawn without replacement, there will be 51 cards left in the deck, out of which 26 are red (13 hearts and 13 diamonds). So, the number of favorable outcomes for the second card being red is 26.

Step 2: Determine the number of total possible outcomes.
Since we're drawing two cards without replacement, each card draw affects the number of remaining cards. For the first card draw, we have a deck of 52 cards, but for the second card draw, we have a deck of 51 cards.

Therefore, the total possible outcomes are the number of ways to choose any two cards from a 52-card deck without replacement. This is calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen.

So, the total possible outcomes can be calculated as: C(52, 2) = 52! / (2!(52-2)!) = 52! / (2!50!) = (52 * 51) / (2 * 1) = 1326.

Step 3: Calculate the probability.
Now that we have determined the number of favorable outcomes (13 for the first card being a heart and 26 for the second card being red) and the total number of possible outcomes (1326), we can calculate the probability.

The probability is given by: favorable outcomes / total possible outcomes.

So, the probability that the first card is a heart and the second card is red can be calculated as: (13/52) * (26/51) = 13/102.

Therefore, the probability is 13/102 or approximately 0.127 or 12.7%.