Question

What is the probability of drawing two kings consecutively from a pack of 52 cards if there are 4 kings in the pack and the first card is replaced before the second is drawn? A. \(1 / 221\) B. \(1 / 104\) C. \(1 / 26\) D. \(1 / 169\) E. None of these

Short answer

D. \(\frac{1}{169}\)

Step-by-step solution

- Understand the Problem

We need to find the probability of drawing two kings consecutively from a standard deck of 52 cards, with the condition that the first card is replaced before the second is drawn.

- Identify Total Outcomes

When drawing a card from a standard deck of 52 cards and then replacing it, the total number of outcomes does not change. So, for each draw, there are 52 possible outcomes.

- Determine Probability of Drawing a King

There are 4 kings in a deck of 52 cards. The probability of drawing a king in one draw is \(\frac{4}{52}\) which simplifies to \(\frac{1}{13}\).

- Determine Probability of Drawing Two Kings

Since the first card is replaced, the probability of drawing a king remains the same for the second draw. Multiply the probabilities of the two independent events: \(\frac{1}{13}\) * \(\frac{1}{13}\) which equals \(\frac{1}{169}\).

- Match the Probability to the Options

The computed probability \(\frac{1}{169}\) matches option D.

Key concepts

Probability Calculations

Probability is a way to measure the chance of an event happening. We calculate probability by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you wanted to find the probability of drawing a king from a deck of 52 cards, you count the number of kings (4) and divide by the total number of cards (52).

This can be expressed as:
\(\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{4}{52} = \frac{1}{13}\)

To find the probability of multiple sequential events, we multiply the probabilities of each individual event. This method assumes the events are independent (meaning each event doesn't affect the others).

In our example, drawing a king, replacing it, and then drawing another king again involves independent events, so:
\( \text{Probability of drawing two kings consecutively} = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169} \)

Independent Events

Independent events are events where the outcome of one event does not affect the outcome of another. In probability calculations, this is a crucial concept because it affects how probabilities are combined.

For instance, when drawing a king from a deck of cards and then replacing it, the condition of replacing the card makes the events independent. The deck remains the same for every draw. Thus:
\( \text{Probability(First King)} = \frac{1}{13}\)
and
\( \text{Probability(Second King)} = \frac{1}{13}\)

Since the events are independent, we multiply their probabilities:
\( \text{Total Probability} = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169}\)

If the replacement did not happen, the events would be dependent. For example, after drawing and retaining a king, there would be 51 cards left, and only 3 of them would be kings for the second draw.

Deck of Cards Probability

Calculating probabilities with a deck of cards is a common exercise in probability questions. A standard deck has 52 cards, with four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, including one king.

Understanding the structure of the deck helps in calculating various probabilities. For example, let’s say you want to find the probability of drawing any face card (jack, queen, or king). There are 12 face cards in a deck (3 per suit). So, the probability is:

\[ \text{Probability(Face card)} = \frac{12}{52} = \frac{3}{13} \]

For our specific problem, the odds of drawing two kings consecutively with replacement means the process resets after each draw:

1. Draw a king: \( \frac{1}{13} \)
2. Replace the card and shuffle.
3. Draw another king: \( \frac{1}{13} \)

Finally, we combine the probabilities by multiplying them:
\[ \text{Probability(Two kings)} = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169} \]