Problem 18 - Thinking Mathematically

Question

You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a 3 each time.

Step-by-Step Solution

Answer

The probability of drawing a '3' twice from a 52-card deck, with each card being replaced after being drawn, is \(\left(\frac{1}{13}\right)^2\).

Step 1: Determine the Total Number of Outcomes

In a standard 52-card deck, there are 52 cards in total. Each card represents a possible outcome. Therefore, there are 52 possible outcomes for each draw.

Step 2: Determine the Number of Successful Outcomes

There are four '3' cards in a standard 52-card deck (3s of hearts, diamonds, clubs, and spades). These represent our successful outcomes.

Step 3: Calculate the Probability of One Draw

The probability of an event is calculated by dividing the number of successful outcomes by the total number of outcomes. Thus, the probability of drawing a '3' from a deck in one draw would be represented by the fraction \(\frac{4}{52}\) or after simplifying this fraction we get \(\frac{1}{13}\).

Step 4: Calculate the Probability of Two Draws

Since both draws are independent, the probability of drawing a '3' in both draws is the square of the result from Step 3. So, we will square \(\frac{1}{13}\) to get \(\left(\frac{1}{13}\right)^2\).

Key Concepts

Independent Events

When we talk about independent events in probability, we mean that the outcome of one event does not affect the other. Each draw from the deck is an independent event because the card's replacement and shuffling ensure no influence on subsequent draws.
For example, if you draw a card and put it back, then shuffle the deck, the second draw is not influenced by the first one. Therefore, the probability of drawing a specific card the first time remains the same the next time, as if no prior draw occurred.
Understanding independent events is crucial since it allows us to calculate joint probabilities by multiplying individual probabilities without worrying about inter-dependencies.

Card Probability

Card probability focuses on calculating the chances of drawing certain cards from a deck. A standard deck of cards has 52 cards, divided into four suits: hearts, diamonds, clubs, and spades.
Each suit contains 13 unique cards, including numbered cards from 2 to 10, plus a jack, queen, king, and ace.

• All cards have an equal chance of being drawn on any single draw if the deck is shuffled well.
• To find the probability of drawing a specific card, you divide the number of successful outcomes by the total number of outcomes.

This concept is fundamental for calculating the probability in card games and exercises like the one given above.

Simple Events

A simple event is an individual outcome or occurrence that cannot be further broken down into simpler components. In the context of probability, each simple event in a probability experiment represents a specific result.
For instance, if you draw a single card from a deck, each specific card draw (e.g., drawing the 3 of hearts) is considered a simple event.
The simplicity comes from the fact that such outcomes are distinct and singular, making them pure simple events. They contribute individually to the overall probability calculation by either happening or not in each trial or experiment.

Replacement in Probability

Replacement in probability plays a significant role in maintaining consistency and independence across trials. When a card is replaced after being drawn, this means putting the card back into the deck after noting down the draw.
By doing a replacement and then reshuffling, each draw is fresh and has no memory of the previous outcomes.

• This practice ensures that the total number of outcomes remains constant (e.g., 52 in a full deck), ensuring fairness in probability calculations.
• Replacement is essential for exercises that require calculations over multiple trials while keeping each event independent as problem conditions dictate.

Understanding the concept of replacement helps in situations where determining unbiased probabilities is crucial, maintaining the integrity of probability experiments.