Smallest value of n | CIE AS Maths | S1 | Binomial Distribution Question
Mathegenius・6 minutes read
A factory producing water pistols has an 8% failure rate, and the probability of at most 2 failures in a sample of 19 is approximately 0.809, while the probability of at least one failure requires a sample size of at least 28 to exceed 0.9. A-level students can utilize logarithmic calculations to determine this, but others may resort to trial and error.
Insights
- The factory producing water pistols has an 8% failure rate, and using the binomial formula, it is determined that there is an 80.9% probability of having at most 2 failures in a sample of 19 pistols, illustrating the effectiveness of statistical methods in assessing product reliability.
- To find the minimum number of pistols needed to ensure that the probability of at least one failure exceeds 90%, calculations show that at least 28 pistols are required, demonstrating the practical application of logarithmic equations and trial-and-error methods in solving real-world problems.
Get key ideas from YouTube videos. It’s free
Recent questions
What is a binomial probability?
Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent trials, each with the same probability of success. It is commonly used in scenarios where there are two possible outcomes, such as success or failure. The formula for calculating binomial probability is \( P(X = r) = \binom{n}{r} p^r q^{n-r} \), where \( n \) is the total number of trials, \( r \) is the number of successes, \( p \) is the probability of success on an individual trial, and \( q \) is the probability of failure (which is \( 1 - p \)). This concept is widely applicable in various fields, including quality control, finance, and healthcare, where understanding the likelihood of certain outcomes is crucial for decision-making.
How do you calculate failure rates?
To calculate failure rates, you first need to determine the total number of items produced and the number of items that failed. The failure rate is then calculated by dividing the number of failures by the total number of items, often expressed as a percentage. For example, if a factory produces 1000 items and 80 of them fail, the failure rate would be \( \frac{80}{1000} \times 100 = 8\% \). This metric is essential for assessing product quality and reliability, allowing manufacturers to identify issues in production processes and implement improvements to reduce the failure rate over time.
What does "at least one failure" mean?
"At least one failure" refers to the occurrence of one or more failures in a given set of trials or tests. In probability terms, it means that the event of interest includes all scenarios where one or more items fail, as opposed to none failing at all. To calculate the probability of at least one failure, it is often easier to first calculate the probability of no failures occurring and then subtract that value from 1. For instance, if the probability of no failures is 0.9, then the probability of at least one failure would be \( 1 - 0.9 = 0.1 \). This concept is particularly important in quality assurance and risk management, where understanding the likelihood of failures can inform better decision-making and preventive measures.
What is trial and error in problem-solving?
Trial and error is a problem-solving method that involves attempting various solutions until the correct one is found. This approach is often used when the solution is not immediately apparent or when there are multiple possible solutions to a problem. In the context of probability and statistics, trial and error can be applied to find values that meet certain conditions, such as determining the minimum number of trials needed to achieve a desired probability of success or failure. While this method can be time-consuming, it is a practical way to explore possibilities and learn from mistakes, making it a valuable tool in both academic and real-world scenarios.
What are logarithms used for in calculations?
Logarithms are mathematical tools used to simplify complex calculations, particularly those involving exponential growth or decay. They are especially useful in scenarios where you need to solve for an exponent in equations. For example, in probability calculations, logarithms can help determine the number of trials needed to achieve a specific probability threshold. By converting multiplicative relationships into additive ones, logarithms make it easier to manipulate equations and find solutions. In practical applications, such as in the context of quality control or risk assessment, logarithms can help analysts quickly assess probabilities and make informed decisions based on statistical data.
Related videos
Mathe by Daniel Jung
Laplace, Laplaceversuch, Laplaceexperiment | Mathe by Daniel Jung
Dane McGuckian (STATSprofessor)
Using a binomial probability table to solve cumulative probability problems
Mathe - simpleclub
Laplace Wahrscheinlichkeit
Mathe by Daniel Jung
Wahrscheinlichkeitsrechnung, Grundlagen, Schraubenproduktion, Stochastik | Mathe by Daniel Jung
Veritasium
The Riddle That Seems Impossible Even If You Know The Answer
Summary
00:00
Water Pistol Failure Probability Analysis
- A factory produces water pistols with an 8% failure rate; in a sample of 19, the probability of at most 2 failures is calculated using the binomial formula, yielding 0.809.
- The binomial formula used is \( P(X = r) = \binom{n}{r} p^r q^{n-r} \), where \( n = 19 \), \( p = 0.08 \), and \( q = 0.92 \).
- For the second problem, the probability that at least one water pistol fails is greater than 0.9, leading to the equation \( 1 - P(X = 0) > 0.9 \).
- Rearranging gives \( 0.92^n < 0.1 \); applying logarithms results in \( n < 27.615 \), while trial and error confirms \( n = 28 \) as the smallest integer satisfying the condition.
- A-level students can use logarithms, while others can apply trial and error to find \( n \); testing values like 20, 25, and 27 approaches the threshold of 0.1.
