King Then Jack: Probability With Card Replacement

Hey guys! Let's dive into a fun probability problem involving Giulia and her deck of cards. She's got a special set of 6 cards: 4 kings, 1 queen, and 1 jack. Giulia's playing a game where she draws a card, puts it back (that's the crucial "with replacement" part!), and then draws another card. Our mission is to figure out the probability of her drawing a king first, and then a jack. This is a classic probability question, and breaking it down step-by-step will make it super clear. So, grab your thinking caps, and let's get started!

Understanding the Basics of Probability

Before we jump into Giulia's game, let’s quickly recap the basics of probability. Probability is simply the chance of something happening. We calculate it by dividing the number of favorable outcomes (the outcomes we're interested in) by the total number of possible outcomes. For instance, if you flip a fair coin, there are two possible outcomes: heads or tails. The probability of getting heads is 1 (favorable outcome) divided by 2 (total outcomes), or 1/2. That means there’s a 50% chance of flipping heads. Now, in Giulia's card game, we're dealing with two events happening one after the other: drawing a king and then drawing a jack. To find the probability of two independent events occurring in sequence, we multiply their individual probabilities. Remember, events are independent if the outcome of one doesn't affect the outcome of the other. The “with replacement” part of Giulia’s game makes the two draws independent because the first card is put back into the deck before the second draw, keeping the total number of cards and the number of each type of card the same for the second draw. So, with these basic concepts in mind, we are ready to tackle the original problem presented by Giulia’s card game.

Calculating the Probability of Drawing a King First

Okay, let's zoom in on the first event: Giulia drawing a king. In her deck of 6 cards, there are 4 kings. So, the number of favorable outcomes (drawing a king) is 4. The total number of possible outcomes is 6, since there are 6 cards in total. Therefore, the probability of Giulia drawing a king on her first draw, which we can denote as P(King), is 4/6. We can simplify this fraction by dividing both the numerator (4) and the denominator (6) by their greatest common divisor, which is 2. This gives us a simplified probability of 2/3. So, there's a 2/3 chance, or roughly a 66.67% chance, that Giulia will draw a king on her first try. Remember, this is just the first part of the problem. We still need to figure out the probability of her drawing a jack on the second draw, and then combine these probabilities to find the overall probability of drawing a king and then a jack. Understanding this initial probability is crucial, as it sets the stage for the next step in our calculation. The simplicity of this calculation underscores the importance of clearly defining the favorable and total possible outcomes.

Calculating the Probability of Drawing a Jack Second (with Replacement)

Now for the second part: Giulia drawing a jack. This is where the "with replacement" part comes into play. Because Giulia puts the first card back into the deck, the deck is exactly the same for the second draw as it was for the first. This means there are still 6 cards in total, and there's still only 1 jack. The probability of drawing a jack, which we’ll call P(Jack), is therefore 1 (favorable outcome – drawing a jack) divided by 6 (total possible outcomes). So, P(Jack) = 1/6. This is a significantly lower probability than drawing a king, which makes sense since there are fewer jacks in the deck than kings. The key thing to remember here is that the replacement ensures that the first draw doesn't impact the probabilities of the second draw. If Giulia hadn't replaced the card, and if she had drawn a king on the first draw, the probabilities for the second draw would be different (there would be only 5 cards left, and potentially fewer kings). However, because of the replacement, the second draw is an independent event, and we can calculate its probability without considering the outcome of the first draw. This independence is crucial for the next step, where we combine these two probabilities.

Combining the Probabilities: King, then Jack

Alright, we've calculated the probability of Giulia drawing a king first (P(King) = 2/3) and the probability of her drawing a jack second (P(Jack) = 1/6). Now, how do we find the probability of both events happening in sequence – drawing a king and then drawing a jack? As we discussed earlier, when we're dealing with independent events (and these are independent because of the replacement), we multiply their individual probabilities. So, the probability of drawing a king and then a jack, which we'll call P(King, then Jack), is P(King) multiplied by P(Jack). That means P(King, then Jack) = (2/3) * (1/6). To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (2/3) * (1/6) = (2 * 1) / (3 * 6) = 2/18. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us a simplified probability of 1/9. Therefore, the probability of Giulia drawing a king and then a jack is 1/9. This means that if Giulia played this game many, many times, we would expect her to draw a king followed by a jack about 1 out of every 9 times.

Final Answer and Key Takeaways

So, to recap, the probability of Giulia drawing a king and then a jack from her deck of cards with replacement is 1/9. That's our final answer! This problem beautifully illustrates the fundamental principles of probability, especially how to calculate the probability of independent events occurring in sequence. The key takeaway here is the importance of understanding the concept of “with replacement.” It ensures that each draw is independent, simplifying the calculations. If the cards weren't replaced, the problem would become more complex, as the outcome of the first draw would influence the probabilities of the second draw. Remember, when faced with probability problems, always break them down into smaller, manageable steps. Identify the individual probabilities, and then use the appropriate rules (like multiplying for independent events) to combine them. With a little practice, you'll become a probability pro in no time! Hope this helped, and happy calculating!