A Set Of 3 Cards, Spelling The Word ADD, Are Placed Face Down On The Table. Determine $P(A, A$\] If Two Cards Are Randomly Selected With Replacement.A. $\frac{1}{3}$ B. $\frac{1}{9}$ C. $\frac{2}{3}$ D.
Introduction
In probability theory, the concept of drawing cards with replacement is a fundamental idea. When cards are drawn with replacement, the probability of drawing a specific card remains the same for each draw. In this article, we will explore the probability of drawing two A's from a set of three cards, spelling the word "ADD," when the cards are randomly selected with replacement.
Understanding the Problem
We have a set of three cards, each spelling a letter of the word "ADD." The cards are placed face down on the table, and we are asked to determine the probability of drawing two A's when two cards are randomly selected with replacement.
Defining the Sample Space
To solve this problem, we need to define the sample space, which is the set of all possible outcomes. In this case, the sample space consists of the following outcomes:
- AA (drawing two A's)
- AD (drawing an A and a D)
- DA (drawing a D and an A)
- DD (drawing two D's)
Calculating the Probability
Since the cards are drawn with replacement, the probability of drawing an A remains the same for each draw. Let's calculate the probability of drawing two A's.
The probability of drawing an A on the first draw is 1/3, since there is one A out of three cards.
The probability of drawing an A on the second draw is also 1/3, since the cards are replaced, and the probability remains the same.
To calculate the probability of drawing two A's, we multiply the probabilities of each draw:
P(AA) = P(A) × P(A) = (1/3) × (1/3) = 1/9
Conclusion
In conclusion, the probability of drawing two A's from a set of three cards, spelling the word "ADD," when the cards are randomly selected with replacement is 1/9.
Answer
The correct answer is B. .
Additional Examples
To further illustrate the concept of drawing cards with replacement, let's consider a few additional examples:
- What is the probability of drawing two A's from a set of four cards, spelling the word "ABCD," when the cards are randomly selected with replacement?
- What is the probability of drawing two D's from a set of three cards, spelling the word "ADD," when the cards are randomly selected with replacement?
Solutions
- The probability of drawing two A's from a set of four cards, spelling the word "ABCD," when the cards are randomly selected with replacement is 1/16.
- The probability of drawing two D's from a set of three cards, spelling the word "ADD," when the cards are randomly selected with replacement is 1/9.
Final Thoughts
Introduction
In our previous article, we explored the concept of drawing cards with replacement and calculated the probability of drawing two A's from a set of three cards, spelling the word "ADD." In this article, we will answer some frequently asked questions (FAQs) related to probability of drawing cards with replacement.
Q&A
Q: What is the probability of drawing a specific card from a deck of cards when the cards are drawn with replacement?
A: The probability of drawing a specific card from a deck of cards when the cards are drawn with replacement is 1/n, where n is the total number of cards in the deck.
Q: What is the probability of drawing two specific cards from a deck of cards when the cards are drawn with replacement?
A: The probability of drawing two specific cards from a deck of cards when the cards are drawn with replacement is (1/n) × (1/n) = 1/n^2, where n is the total number of cards in the deck.
Q: What is the probability of drawing a specific card from a set of cards when the cards are drawn without replacement?
A: The probability of drawing a specific card from a set of cards when the cards are drawn without replacement is 1/(n-1), where n is the total number of cards in the set.
Q: What is the probability of drawing two specific cards from a set of cards when the cards are drawn without replacement?
A: The probability of drawing two specific cards from a set of cards when the cards are drawn without replacement is (1/(n-1)) × (1/(n-2)), where n is the total number of cards in the set.
Q: Can you provide an example of drawing cards with replacement?
A: Let's say we have a deck of 52 cards, and we want to draw two cards with replacement. The probability of drawing a specific card on the first draw is 1/52. Since the cards are replaced, the probability of drawing the same card on the second draw is also 1/52. Therefore, the probability of drawing two specific cards is (1/52) × (1/52) = 1/2704.
Q: Can you provide an example of drawing cards without replacement?
A: Let's say we have a deck of 52 cards, and we want to draw two cards without replacement. The probability of drawing a specific card on the first draw is 1/52. Since the cards are not replaced, the probability of drawing the same card on the second draw is 1/51. Therefore, the probability of drawing two specific cards is (1/52) × (1/51) = 1/2652.
Q: What is the difference between drawing cards with replacement and drawing cards without replacement?
A: The main difference between drawing cards with replacement and drawing cards without replacement is that the probability of drawing a specific card changes when the cards are drawn without replacement. When the cards are drawn with replacement, the probability of drawing a specific card remains the same for each draw.
Q: Can you provide a real-world example of drawing cards with replacement?
A: Let's say we are playing a game where we draw cards from a deck with replacement. Each card has a point value associated with it, and we want to calculate the probability of drawing a specific card with a certain point value. In this case, drawing cards with replacement is a good model to use, as the probability of drawing a specific card remains the same for each draw.
Q: Can you provide a real-world example of drawing cards without replacement?
A: Let's say we are playing a game where we draw cards from a deck without replacement. Each card has a point value associated with it, and we want to calculate the probability of drawing a specific card with a certain point value. In this case, drawing cards without replacement is a good model to use, as the probability of drawing a specific card changes after each draw.
Conclusion
In conclusion, drawing cards with replacement and drawing cards without replacement are two different concepts in probability theory. By understanding the difference between these two concepts, we can apply them to real-world problems and make informed decisions. The examples provided in this article demonstrate the application of these concepts to real-world problems.