Suppose that two cards are randomly selected from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a space if the sampling is done without replacement?

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To determine the probability, we need to calculate two probabilities separately:

1. Probability of drawing a spade on the first card.
2. Probability of drawing another spade on the second card, given that the first card was a spade.

Step 1: Probability of drawing a spade on the first card.
In a standard 52-card deck, there are 13 spades out of 52 cards.
So, the probability of drawing a spade on the first card is 13/52, which simplifies to 1/4.

Step 2: Probability of drawing another spade on the second card, given that the first card was a spade.
After the first card is drawn, there are now 51 cards remaining in the deck and 12 spades left.
So, the probability of drawing a spade on the second card, given that the first card was a spade, is 12/51.

To find the overall probability, we multiply the probabilities from step 1 and step 2, since both events need to occur:

(1/4) * (12/51) = 12/204 = 3/51 = 1/17.

Therefore, the probability that the first card is a spade and the second card is also a spade, when the sampling is done without replacement, is 1/17.

To find the probability that the first card is a spade and the second card is also a spade, we can use the concept of conditional probability.

Step 1: Determine the total number of ways the two cards can be selected without replacement.
Since there are 52 cards in a deck, there are 52 choices for the first card. After the first card is selected, there are 51 remaining cards for the second card. Therefore, the total number of ways to select two cards without replacement is 52 * 51.

Step 2: Determine the number of favorable outcomes, i.e., the number of ways to select a spade for the first card and another spade for the second card.
There are 13 spades in a deck (because each suit has 13 cards). So, for the first card to be a spade, there are 13 choices. After one spade is selected, there are 12 spades remaining for the second card. Therefore, the number of favorable outcomes is 13 * 12.

Step 3: Calculate the probability using the formula: probability = favorable outcomes / total outcomes.
In this case, the probability would be (13 * 12) / (52 * 51).

To simplify this, you can cancel out common factors:
(13 * 12) / (52 * 51) = (13 / 52) * (12 / 51) = (1 / 4) * (4 / 17) = 1 / 17.

So, the probability that the first card is a spade and the second card is also a spade, when sampled without replacement, is 1/17.

The probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

13/52 * (13-1)/(52-1) = ?