Question

Two cards are drawn from a pack of cards one after another so that the first card is replaced before drawing the second card. What is the probability that the first card is an ace and the second is a number card? (1) \(\frac{9}{169}\) (2) \(\frac{1}{52}\) (3) \(\frac{1}{4}\) (4) \(\frac{17}{52}\)

Short answer

Answer: The probability is \(\frac{9}{169}\).

Step-by-step solution

Identify the Probability of Drawing an Ace

There are 4 aces in a standard deck of 52 cards. To find the probability of drawing an ace, we can simply divide the number of aces by the total number of cards: P(Ace) = \(\frac{4}{52}\)

Identify the Probability of Drawing a Number Card

There are 36 number cards in a standard deck of 52 cards (2-10 of each suit). To find the probability of drawing a number card, we can simply divide the number of number cards by the total number of cards: P(Number) = \(\frac{36}{52}\)

Find the Probability of Both Events Happening

Since the first card is replaced before drawing the second card, the two events are independent. Therefore, we can multiply the probabilities of each event happening to find the probability of both events happening: P(Ace and Number) = P(Ace) * P(Number) = \(\frac{4}{52}\) * \(\frac{36}{52}\)

Simplify the Result

Now we can simplify the result to find the final probability: P(Ace and Number) = \(\frac{4*36}{52*52}\) = \(\frac{144}{2704}\) = \(\boxed{\frac{9}{169}}\) So the correct answer is (1) \(\frac{9}{169}\).

Key concepts

Independent Events

When evaluating the probability of scenarios involving multiple events, it is essential to understand whether these events are independent.
Independent events are those for which the outcome of one event does not affect the outcome of another. This is a crucial concept when calculating combined probabilities.
For example, drawing two cards from a deck with replacement ensures that the events are independent.

• The probability of drawing a specific card remains constant because the deck is reset back to its full 52 cards after each draw.
• This independence allows you to multiply their probabilities to find the joint probability of both events happening.

Understanding independent events allows for accurate and straightforward probability calculations in scenarios involving multiple actions.

Deck of Cards

A standard deck of cards is fundamental to many probability exercises and games.
Understanding its structure is key to solving problems related to it. A standard deck contains 52 cards in total, divided into four suits: hearts, diamonds, clubs, and spades.

• Each suit has 13 cards, including one Ace, face cards (Jack, Queen, King), and number cards numbered from 2 to 10.
• The face and number cards add up to make significant components when calculating probabilities associated with the deck.

In the exercise, the concept of a deck is utilized to determine how many favorable outcomes (number cards, aces) there are out of the total number of cards to set up ratio-based probability calculations.
Knowing your deck is vital when working through these exercises as it provides a solid foundation for approaching complex probability problems.

Probability of Drawing Cards

The art of calculating probabilities often hinges on understanding the favorable outcomes divided by the total possible outcomes.
This is simplified using a deck of cards due to its regular structure.

• When calculating the probability of a single event, such as drawing an ace from a deck, count the favorable outcomes (aces) and divide by the total number (52 cards).
• Similarly, to find the probability of drawing a number card (cards numbered 2-10), you would count all such cards and divide by 52.

In sequences involving more than one card draw, such as drawing an ace and then a number card, probabilities are multiplied when the events are independent, like in a situation with replacement.
Hence the total probability is a product of the independent probabilities, providing the chance of both scenarios happening together.