Clarence Bread Quant Question
Introduction
Clarence is getting this bread, literally. He has 7 small loaves and 3 large loaves of bread in a bag. He draws in his bag and pulls out a loaf of bread uniformly at random, one-by-one. If it is a small loaf, he puts it back in the bag and draws again. If it is a large loaf, he stops drawing. What is the probability that Clarence will stop drawing after exactly 5 draws?
Understanding the Problem
To tackle this problem, we need to understand the underlying probability distribution. Since Clarence is drawing loaves uniformly at random, the probability of drawing a small loaf is 7/10, and the probability of drawing a large loaf is 3/10. We are interested in finding the probability that Clarence will stop drawing after exactly 5 draws, which means he must draw 4 small loaves and 1 large loaf in the first 5 draws.
Breaking Down the Problem
Let's break down the problem into smaller, manageable parts. We can use the concept of conditional probability to solve this problem. The probability of drawing a large loaf on the 5th draw, given that the first 4 draws are small loaves, is 3/10. The probability of drawing a small loaf on the 5th draw, given that the first 4 draws are small loaves, is 7/10.
Calculating the Probability
To calculate the probability that Clarence will stop drawing after exactly 5 draws, we need to consider all possible combinations of small and large loaves in the first 5 draws. We can use the concept of combinations to calculate the number of ways to choose 4 small loaves and 1 large loaf in 5 draws.
The number of ways to choose 4 small loaves and 1 large loaf in 5 draws is given by the combination formula:
C(5, 4) = 5! / (4! * (5-4)!)
C(5, 4) = 5! / (4! * 1!)
C(5, 4) = 5
There are 5 ways to choose 4 small loaves and 1 large loaf in 5 draws.
Calculating the Probability of Each Combination
Now that we have the number of ways to choose 4 small loaves and 1 large loaf in 5 draws, we need to calculate the probability of each combination. The probability of drawing a small loaf on each of the first 4 draws is (7/10)^4, and the probability of drawing a large loaf on the 5th draw is (3/10).
The probability of each combination is given by:
P(combination) = (7/10)^4 * (3/10)
P(combination) = (2401/100000) * (3/10)
P(combination) = 7203/100000
Calculating the Total Probability
Now that we have the probability of each combination, we can calculate the total probability by summing up the probabilities of all combinations.
The total probability is given by:
P(total) = P(combination 1) + P(combination 2) + ... + P(combination 5)
P(total) = 7203/100000 + 7203/100000 + ... + 7203/100000
P(total) = 5 * 7203/100000
P(total) = 36015/100000
P(total) = 0.36015
Conclusion
In this article, we explored the problem of Clarence drawing loaves of bread uniformly at random from a bag. We used the concept of conditional probability and combinations to calculate the probability that Clarence will stop drawing after exactly 5 draws. We found that the probability is approximately 0.36015.
Probability and Combinatorics: A Delightful Exploration
Probability and combinatorics are fascinating fields that have numerous applications in real-life scenarios. In this article, we explored a delightful problem that involves probability and combinatorics. We used the concept of conditional probability and combinations to calculate the probability that Clarence will stop drawing after exactly 5 draws.
Real-Life Applications
Probability and combinatorics have numerous real-life applications. For example, in insurance, probability and combinatorics are used to calculate the probability of an event occurring, such as a car accident. In finance, probability and combinatorics are used to calculate the probability of a stock price moving in a certain direction.
Conclusion
In conclusion, probability and combinatorics are fascinating fields that have numerous applications in real-life scenarios. In this article, we explored a delightful problem that involves probability and combinatorics. We used the concept of conditional probability and combinations to calculate the probability that Clarence will stop drawing after exactly 5 draws. We found that the probability is approximately 0.36015.
References
- [1] "Probability and Combinatorics" by David M. Bressoud
- [2] "A First Course in Probability" by Sheldon M. Ross
- [3] "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
Glossary
- Conditional Probability: The probability of an event occurring given that another event has occurred.
- Combinations: The number of ways to choose a certain number of items from a larger set, without regard to order.
- Uniformly at Random: A random variable that takes on values from a given set with equal probability.
Clarence Bread Quant Question: A Delightful Exploration of Probability and Combinatorics - Q&A ===========================================================
Introduction
In our previous article, we explored the problem of Clarence drawing loaves of bread uniformly at random from a bag. We used the concept of conditional probability and combinations to calculate the probability that Clarence will stop drawing after exactly 5 draws. In this article, we will answer some frequently asked questions related to the problem.
Q&A
Q: What is the probability that Clarence will stop drawing after exactly 6 draws?
A: To calculate the probability that Clarence will stop drawing after exactly 6 draws, we need to consider all possible combinations of small and large loaves in the first 6 draws. We can use the concept of combinations to calculate the number of ways to choose 5 small loaves and 1 large loaf in 6 draws.
The number of ways to choose 5 small loaves and 1 large loaf in 6 draws is given by the combination formula:
C(6, 5) = 6! / (5! * (6-5)!)
C(6, 5) = 6! / (5! * 1!)
C(6, 5) = 6
There are 6 ways to choose 5 small loaves and 1 large loaf in 6 draws.
The probability of each combination is given by:
P(combination) = (7/10)^5 * (3/10)
P(combination) = (16807/1000000) * (3/10)
P(combination) = 50421/10000000
The total probability is given by:
P(total) = P(combination 1) + P(combination 2) + ... + P(combination 6)
P(total) = 50421/10000000 + 50421/10000000 + ... + 50421/10000000
P(total) = 6 * 50421/10000000
P(total) = 302526/10000000
P(total) = 0.0302526
Therefore, the probability that Clarence will stop drawing after exactly 6 draws is approximately 0.0302526.
Q: What is the probability that Clarence will stop drawing after exactly 7 draws?
A: To calculate the probability that Clarence will stop drawing after exactly 7 draws, we need to consider all possible combinations of small and large loaves in the first 7 draws. We can use the concept of combinations to calculate the number of ways to choose 6 small loaves and 1 large loaf in 7 draws.
The number of ways to choose 6 small loaves and 1 large loaf in 7 draws is given by the combination formula:
C(7, 6) = 7! / (6! * (7-6)!)
C(7, 6) = 7! / (6! * 1!)
C(7, 6) = 7
There are 7 ways to choose 6 small loaves and 1 large loaf in 7 draws.
The probability of each combination is given by:
P(combination) = (7/10)^6 * (3/10)
P(combination) = (117649/10000000) * (3/10)
P(combination) = 353947/100000000
The total probability is given by:
P(total) = P(combination 1) + P(combination 2) + ... + P(combination 7)
P(total) = 353947/100000000 + 353947/100000000 + ... + 353947/100000000
P(total) = 7 * 353947/100000000
P(total) = 2481639/100000000
P(total) = 0.02481639
Therefore, the probability that Clarence will stop drawing after exactly 7 draws is approximately 0.02481639.
Q: What is the probability that Clarence will stop drawing after exactly 8 draws?
A: To calculate the probability that Clarence will stop drawing after exactly 8 draws, we need to consider all possible combinations of small and large loaves in the first 8 draws. We can use the concept of combinations to calculate the number of ways to choose 7 small loaves and 1 large loaf in 8 draws.
The number of ways to choose 7 small loaves and 1 large loaf in 8 draws is given by the combination formula:
C(8, 7) = 8! / (7! * (8-7)!)
C(8, 7) = 8! / (7! * 1!)
C(8, 7) = 8
There are 8 ways to choose 7 small loaves and 1 large loaf in 8 draws.
The probability of each combination is given by:
P(combination) = (7/10)^7 * (3/10)
P(combination) = (823543/100000000) * (3/10)
P(combination) = 2470629/1000000000
The total probability is given by:
P(total) = P(combination 1) + P(combination 2) + ... + P(combination 8)
P(total) = 2470629/1000000000 + 2470629/1000000000 + ... + 2470629/1000000000
P(total) = 8 * 2470629/1000000000
P(total) = 19765032/1000000000
P(total) = 0.019765032
Therefore, the probability that Clarence will stop drawing after exactly 8 draws is approximately 0.019765032.
Q: What is the probability that Clarence will stop drawing after exactly 9 draws?
A: To calculate the probability that Clarence will stop drawing after exactly 9 draws, we need to consider all possible combinations of small and large loaves in the first 9 draws. We can use the concept of combinations to calculate the number of ways to choose 8 small loaves and 1 large loaf in 9 draws.
The number of ways to choose 8 small loaves and 1 large loaf in 9 draws is given by the combination formula:
C(9, 8) = 9! / (8! * (9-8)!)
C(9, 8) = 9! / (8! * 1!)
C(9, 8) = 9
There are 9 ways to choose 8 small loaves and 1 large loaf in 9 draws.
The probability of each combination is given by:
P(combination) = (7/10)^8 * (3/10)
P(combination) = (5764801/10000000000) * (3/10)
P(combination) = 17294403/100000000000
The total probability is given by:
P(total) = P(combination 1) + P(combination 2) + ... + P(combination 9)
P(total) = 17294403/100000000000 + 17294403/100000000000 + ... + 17294403/100000000000
P(total) = 9 * 17294403/100000000000
P(total) = 155546227/100000000000
P(total) = 0.00155546227
Therefore, the probability that Clarence will stop drawing after exactly 9 draws is approximately 0.00155546227.
Q: What is the probability that Clarence will stop drawing after exactly 10 draws?
A: To calculate the probability that Clarence will stop drawing after exactly 10 draws, we need to consider all possible combinations of small and large loaves in the first 10 draws. We can use the concept of combinations to calculate the number of ways to choose 9 small loaves and 1 large loaf in 10 draws.
The number of ways to choose 9 small loaves and 1 large loaf in 10 draws is given by the combination formula:
C(10, 9) = 10! / (9! * (10-9)!)
C(10, 9) = 10! / (9! * 1!)
C(10, 9) = 10
There are 10 ways to choose 9 small loaves and 1 large loaf in 10 draws.
The probability of each combination is given by:
P(combination) = (7/10)^9 * (3/10)
P(combination) = (387420489/10000000000000) * (3/10)
P(combination) = 1162261474/100000000000000
The total probability is given by:
P(total) = P(combination 1) + P(combination 2) + ... + P(combination 10)
P(total) = 1162261474/100000000000000 + 1162261474/100000000000000 + ... + 1162261474/100000000000000
P(total) = 10 * 1162261474/100000000000000
P(total) = 11622614740/100000000000000
P(total) = 0.0001162261474
Therefore, the probability that Clarence will stop drawing after exactly 10 draws is approximately 0.0001162261474.
Conclusion
In this article, we answered some frequently asked questions related to the problem of Clarence drawing loaves