Chapter 8: Problem 5
What is the relationship between \(b(k ; n, p)\) and \(b(n-k ; n, p)\) when \(p=1 / 2 ?\) Does this relationship hold if \(p \neq 1 / 2 ?\)
Short Answer
Expert verified
The relationship is symmetric at \( p = 1/2 \), but not for \( p \neq 1/2 \).
Step by step solution
01
Understanding the Binomial Distribution
The function \( b(k; n, p) \) represents the binomial probability, which is the probability of getting exactly \( k \) successes in \( n \) independent trials, each with a probability \( p \) of success. It is given by the formula \( b(k; n, p) = \binom{n}{k} p^k (1-p)^{n-k} \).
02
Binomial Distribution with Equal Probability
When \( p = 1/2 \), the binomial probability becomes \( b(k; n, 1/2) = \binom{n}{k} \left( \frac{1}{2} \right)^n \). The expression simplifies to be symmetric about \( k \) because the chosen terms \( \binom{n}{k} \) and \( \binom{n}{n-k} \) are identical.
03
Establishing the Symmetry Relationship
From the properties of binomial coefficients, we know that \( \binom{n}{k} = \binom{n}{n-k} \). Therefore, substitute these binomial coefficients in: \[ b(k; n, 1/2) = b(n-k; n, 1/2). \] This shows that the relationship is symmetric around \( p = 1/2 \).
04
Checking the Case When \( p \neq 1/2 \)
For \( p eq 1/2 \), the symmetry does not hold. The expression \( p^k (1-p)^{n-k} \) is not equal to \( p^{n-k} (1-p)^k \), which means \[ b(k; n, p) eq b(n-k; n, p) \] if \( p eq 1/2 \). The lack of symmetry results from the unequal contribution of \( p^k \) compared to \( (1-p)^{n-k} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Coefficients
Binomial coefficients are an essential part of the binomial distribution formula. They are represented by the symbol \( \binom{n}{k} \), and used to count the number of ways to choose \( k \) successes out of \( n \) trials. Imagine selecting a committee from a group: binomial coefficients help determine all possible selections.
Using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), you calculate these coefficients. Here, \( n! \) denotes a factorial, which is the product of all positive integers up to \( n \).
The magic happens because of their mathematical properties. One of these properties is symmetry. This symmetry means \( \binom{n}{k} = \binom{n}{n-k} \). Hence, choosing \( k \) successes in \( n \) trials is the same as choosing \( n-k \) failures. This property plays a vital role in proving why certain binomial distributions are symmetric about their mean.
Understanding binomial coefficients is crucial in probability theory because they lay the foundation for various binomial distributions.
Using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), you calculate these coefficients. Here, \( n! \) denotes a factorial, which is the product of all positive integers up to \( n \).
The magic happens because of their mathematical properties. One of these properties is symmetry. This symmetry means \( \binom{n}{k} = \binom{n}{n-k} \). Hence, choosing \( k \) successes in \( n \) trials is the same as choosing \( n-k \) failures. This property plays a vital role in proving why certain binomial distributions are symmetric about their mean.
Understanding binomial coefficients is crucial in probability theory because they lay the foundation for various binomial distributions.
Symmetry in Probability
Symmetry in probability, particularly with binomial distributions, highlights the balance of outcomes. When the probability of success and failure is equal, such as \( p = 1/2 \), the distribution becomes symmetric. This means getting exactly \( k \) successes can be mirrored by the same chance of getting \( n-k \) failures.
The symmetry property of binomial coefficients, \( \binom{n}{k} = \binom{n}{n-k} \), reinforces this concept. Consequently, when each trial's success or failure probability is equal, the outcome probabilities reflect this balance.
For example, flipping a fair coin is a typical scenario where symmetry in probability is evident. The chance to get two heads in four flips is the same as getting two tails, due to symmetry. This balanced probability outcome forms the reason why we can make certain assumptions and predictions about these types of events in probability theory.
However, this symmetry only holds true with \( p = 1/2 \). Once we have \( p eq 1/2 \), the equality no longer holds, and the symmetry is broken. This shift showcases the impact of unequal probability of success and failure on the distribution's symmetry.
The symmetry property of binomial coefficients, \( \binom{n}{k} = \binom{n}{n-k} \), reinforces this concept. Consequently, when each trial's success or failure probability is equal, the outcome probabilities reflect this balance.
For example, flipping a fair coin is a typical scenario where symmetry in probability is evident. The chance to get two heads in four flips is the same as getting two tails, due to symmetry. This balanced probability outcome forms the reason why we can make certain assumptions and predictions about these types of events in probability theory.
However, this symmetry only holds true with \( p = 1/2 \). Once we have \( p eq 1/2 \), the equality no longer holds, and the symmetry is broken. This shift showcases the impact of unequal probability of success and failure on the distribution's symmetry.
Probability Theory
Probability theory forms the backbone of many statistical calculations, focusing on the likelihood of different outcomes. In the context of binomial distribution, it determines how likely it is to achieve a certain number of successes in a given number of trials.
The binomial distribution function \( b(k; n, p) \) demonstrates this theory in action, calculating probabilities for each potential number of successes. Each trial is independent, meaning previous outcomes do not affect future ones. This independence is a key feature in probability theory and ensures accurate predictions regarding random events.
A fundamental aspect of probability theory is understanding variations in expected outcomes with changes in probability, \( p \). When \( p = 1/2 \), there is symmetry in outcomes, but different probabilities, \( p eq 1/2 \), result in asymmetric distributions.
Probability theory not only helps us calculate expected outcomes but also comprehend how varied conditions shift these expectations. It's the reason financial analysts predict market trends or why meteorologists forecast weather patterns; it's all about understanding the probabilities of different outcomes in a structured manner.
The binomial distribution function \( b(k; n, p) \) demonstrates this theory in action, calculating probabilities for each potential number of successes. Each trial is independent, meaning previous outcomes do not affect future ones. This independence is a key feature in probability theory and ensures accurate predictions regarding random events.
A fundamental aspect of probability theory is understanding variations in expected outcomes with changes in probability, \( p \). When \( p = 1/2 \), there is symmetry in outcomes, but different probabilities, \( p eq 1/2 \), result in asymmetric distributions.
Probability theory not only helps us calculate expected outcomes but also comprehend how varied conditions shift these expectations. It's the reason financial analysts predict market trends or why meteorologists forecast weather patterns; it's all about understanding the probabilities of different outcomes in a structured manner.
