Suppose that two cards are randomly selected from a standard 52-card deck.
b. What is the probability that the first card is a king and the second card is a king if the sampling is done with replacement?

Verified step by step guidance

1. Understand the problem: We are selecting two cards from a standard 52-card deck with replacement, and we want the probability that both cards drawn are kings.
2. Recall that a standard deck has 4 kings out of 52 cards. The probability of drawing a king on the first draw is therefore \(\frac{4}{52}\).
3. Since the sampling is done with replacement, after drawing the first card, we put it back into the deck. This means the deck is restored to 52 cards with 4 kings before the second draw.
4. The probability of drawing a king on the second draw remains \(\frac{4}{52}\) because the deck composition is unchanged due to replacement.
5. To find the combined probability of both events happening (first card is a king AND second card is a king), multiply the probabilities of each independent event: \(\frac{4}{52} \times \frac{4}{52}\).

Key Concepts

Probability with Replacement

When sampling with replacement, each selection is independent because the item is returned to the population before the next draw. This means the probabilities remain constant across draws, simplifying calculations since the total number of possible outcomes does not change.

Calculating Probability of Sequential Events

The probability of multiple independent events occurring in sequence is found by multiplying the probabilities of each individual event. For example, the probability of drawing a king first and then a king again with replacement is the product of the probability of drawing a king each time.

Standard Deck Composition

A standard deck has 52 cards with 4 kings. Knowing the number of favorable outcomes (kings) and total outcomes (52 cards) is essential to calculate the probability of drawing a king on any single draw.