Directions: Suppose you have a standard deck 52 cards (4 suits: hearts, diamonds, clubs, and spades each suit has 13 cards (1/4 of the deck) and those 13 cards include one ace,and numbers 2 through 10 and 3 face cards: a jack a queen and king) suppose you pick a card at randome and then without replacement the first card you pick another card randomly.

Question:
A) what' is the probability that the second card you draw is a heart given that your first draw was heart?
B) what is the probability that your second card is heart in general regardless of the card you drew first?

If you have a heart the first time then there are only 12 hearts left out of 51 cards

To find the probabilities in both scenarios, we need to understand the concept of conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred.

A) To find the probability that the second card drawn is a heart, given that the first card drawn is a heart, we have the following information:

- There are 52 cards in the deck initially.
- There are 13 hearts in the deck initially.
- After drawing the first heart, there are 51 cards left in the deck, and 12 hearts remaining.

The probability can be calculated as follows:

P(Second card is a heart | First card is a heart) = Number of favorable outcomes / Number of possible outcomes

Number of favorable outcomes: 12 hearts remaining in the deck
Number of possible outcomes: 51 cards remaining in the deck

Therefore, the probability that the second card drawn is a heart, given that the first card drawn is a heart, is 12/51.

B) To find the probability that the second card drawn is a heart, regardless of the first card drawn, we have the information mentioned above:

- There are 52 cards in the deck initially.
- There are 13 hearts in the deck initially.

The probability can be calculated as follows:

P(Second card is a heart) = Number of favorable outcomes / Number of possible outcomes

Number of favorable outcomes: 13 hearts in the deck
Number of possible outcomes: 52 cards in the deck

Therefore, the probability that the second card drawn is a heart, regardless of the first card drawn, is 13/52 or 1/4.

In summary:
A) The probability of drawing a second heart, given that the first card drawn is a heart, is 12/51.
B) The probability of drawing a second heart, regardless of the first card drawn, is 13/52 or 1/4.

A) To find the probability that the second card you draw is a heart given that your first draw was a heart, we need to consider the information from the first draw.

Step 1: Calculate the probability of drawing a heart on the first pick.
Since there are 52 cards in the deck and 13 hearts, the probability of drawing a heart on the first pick is 13/52.

Step 2: Calculate the number of cards remaining after the first pick.
After the first pick, there will be 51 cards remaining in the deck, with 12 hearts remaining.

Step 3: Calculate the probability of drawing a heart on the second pick, given that the first card picked was a heart.
Since there are now 12 hearts out of the remaining 51 cards, the probability of drawing a heart on the second pick is 12/51.

Therefore, the probability that the second card you draw is a heart, given that your first draw was a heart, is 12/51.

B) To find the probability that your second card is a heart in general, regardless of the card you drew first, we need to consider the total number of hearts in the deck and the remaining cards after the first pick.

Step 1: Calculate the probability of drawing a heart on the first pick.
Since there are 52 cards in the deck and 13 hearts, the probability of drawing a heart on the first pick is 13/52.

Step 2: Calculate the number of cards remaining after the first pick.
After the first pick, there will be 51 cards remaining in the deck, with 13 hearts remaining.

Step 3: Calculate the probability of drawing a heart on the second pick, regardless of the first card picked.
Since there are still 13 hearts out of the remaining 51 cards, the probability of drawing a heart on the second pick is 13/51.

Therefore, the probability that your second card is a heart in general, regardless of the card you drew first, is 13/51.