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7.10 The Binomial Distribution

2 min read•june 18, 2024

helps us calculate the chances of specific outcomes in experiments with fixed and two possible results. It's like predicting how many times you'll roll a six in ten dice throws.

The is key, using factors like total trials, desired successes, and individual . We can use this to figure out things like the likelihood of getting three heads when flipping a coin five times.

Binomial Probability

Binomial formula for probabilities

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  • Calculates the probability of a specific number of successes in a fixed number of independent trials
  • Formula: P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k} (also known as the for discrete probability distributions)
    • nn: total number of trials (fixed)
    • kk: number of successes desired
    • pp: probability of success on a single trial
    • 1−p1-p: probability of failure on a single trial
  • To use the formula:
    1. Confirm the experiment is binomial (fixed trials, two outcomes, constant probability, )
    2. Identify nn, kk, and pp based on the problem
    3. Calculate the combination (nk)\binom{n}{k} using n!k!(n−k)!\frac{n!}{k!(n-k)!}
    4. Plug values into the formula and simplify
  • Examples:
    • Probability of getting exactly 3 heads in 5 coin flips (fair coin)
    • Probability of 2 defective items in a batch of 10 (10% defect rate)

Probability functions for binomial experiments

  • () gives the probability of each possible outcome
    • Construct by calculating P(X=k)P(X=k) for each kk from 0 to nn using the binomial formula
    • All probabilities in a PDF sum to 1
    • Example: PDF for number of heads in 3 coin flips (0, 1, 2, or 3 heads possible)
  • () gives the probability of kk or fewer successes
    • Construct by summing probabilities from the PDF for each kk from 0 to the desired value
    • CDF starts at 0 and ends at 1
    • Example: CDF for 2 or fewer defective items in a batch of 10

Binomial Experiments

Criteria of binomial experiments

  • Fixed number of trials (nn)
  • Only two possible outcomes per trial (success or failure)
  • Constant probability of success (pp) for all trials
  • Trials are independent of each other (independence ensures that the outcome of one trial does not affect the others)
  • Examples of :
    • Flipping a coin 20 times and counting heads (two outcomes, fixed trials, constant pp, independence)
    • Testing 15 batteries and counting defects (pass/fail, fixed trials, constant defect rate, independence)
  • Non-examples:
    • Drawing cards without replacement (probability changes each draw, violating constant pp)
    • Survey with multiple choice answers (more than two outcomes per trial)

Properties of Binomial Distribution

  • Mean (): μ=np\mu = np
  • : σ2=np(1−p)\sigma^2 = np(1-p), which measures the spread of the distribution
  • The is a , meaning it deals with countable, distinct outcomes

Key Terms to Review (25)

Bernoulli: Bernoulli refers to the principle that describes the behavior of fluid dynamics and probability, particularly in relation to the binomial distribution. In probability, Bernoulli trials are a sequence of experiments where each experiment has two possible outcomes: success or failure. This principle is foundational for understanding the binomial distribution, as it deals with scenarios where there are repeated independent trials with the same probability of success.
Bernoulli trial: A Bernoulli trial is a random experiment that has exactly two possible outcomes: 'success' and 'failure'. Each trial is independent, meaning the outcome of one trial does not affect the outcome of another. This concept is foundational in understanding the binomial distribution, as it allows us to model situations where we are interested in the number of successes in a fixed number of trials.
Binomial Coefficient: The binomial coefficient, often represented as $$C(n, k)$$ or $$\binom{n}{k}$$, counts the number of ways to choose a subset of size $$k$$ from a larger set of size $$n$$ without regard to the order of selection. This concept is crucial in combinatorics and probability, as it helps in calculating probabilities in scenarios where there are two possible outcomes, like success or failure, which is central to the study of distributions and graph structures.
Binomial distribution: A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. This concept is key in understanding how outcomes can be modeled when there are only two possible results, such as success or failure, which connects to concepts like odds and expected value. It also provides a foundation for approximating distributions that resemble normal distributions under certain conditions.
Binomial experiments: A binomial experiment is a statistical experiment that has exactly two possible outcomes for each trial, commonly referred to as 'success' and 'failure.' The trials are independent, and the probability of success remains constant throughout the experiment.
Binomial formula: The binomial formula is a mathematical expression that provides a way to calculate the coefficients of the terms in the expansion of a binomial raised to a positive integer power. This formula is significant because it forms the foundation for understanding binomial distributions, which describe the number of successes in a fixed number of independent Bernoulli trials. The binomial formula helps in determining probabilities and expectations in various real-world situations, making it a crucial tool in statistics and probability theory.
Binomial probability: Binomial probability refers to the likelihood of achieving a specific number of successes in a fixed number of independent trials, each with the same probability of success. This concept is fundamental in the context of the binomial distribution, which models situations where there are two possible outcomes, such as success or failure. The binomial probability formula allows for the calculation of probabilities for various scenarios within these trials, making it a key tool in statistics.
Binomial trials: Binomial trials are a series of experiments where each trial has two possible outcomes: success or failure. They are characterized by a fixed number of trials, independent events, and constant probability of success.
CDF: CDF stands for Cumulative Distribution Function, which is a function that describes the probability that a random variable takes on a value less than or equal to a specific value. In the context of probability distributions, including the binomial distribution, the CDF provides a way to determine the likelihood of obtaining a certain number of successes in a fixed number of trials. It is essential for understanding how probabilities accumulate across different outcomes.
Cumulative distribution function: The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes on a value less than or equal to a specified number. It is a non-decreasing function that ranges from 0 to 1.
Cumulative Distribution Function: A cumulative distribution function (CDF) describes the probability that a random variable takes on a value less than or equal to a specific value. It provides a complete description of the distribution of a random variable, and is crucial in understanding both discrete and continuous probability distributions, showing how probabilities accumulate over the range of possible outcomes.
Discrete Probability Distribution: A discrete probability distribution is a statistical function that describes the likelihood of each possible outcome of a discrete random variable. Each outcome is assigned a probability, and the sum of all probabilities in the distribution must equal one. This concept is crucial for understanding various types of random processes, including those modeled by specific distributions like the binomial distribution.
Expected value: Expected value is a fundamental concept in probability that represents the average outcome of a random event over a large number of trials. It is calculated by multiplying each possible outcome by its probability and summing the results.
Independence: Independence refers to the scenario where the occurrence of one event does not affect the probability of another event occurring. This concept is crucial as it underlies many basic principles in probability, influencing how we calculate probabilities of combined events and affecting distributions such as the binomial distribution.
N choose k: The term 'n choose k' refers to the mathematical notation $$C(n, k)$$ or $$\binom{n}{k}$$, which represents the number of ways to select 'k' items from a total of 'n' items without regard to the order of selection. This concept is fundamental in combinatorics and plays a crucial role in understanding probabilities, particularly in relation to the binomial distribution, which models the number of successes in a series of independent trials.
Pascal: Pascal refers to a mathematical concept named after Blaise Pascal, which is essential for understanding the binomial distribution. It involves a triangular arrangement of coefficients known as Pascal's Triangle, where each number is the sum of the two directly above it. This triangle provides a way to compute combinations, which are fundamental to determining the probabilities associated with the binomial distribution.
PDF: In statistics, PDF stands for Probability Density Function, which describes the likelihood of a continuous random variable taking on a specific value. The PDF is essential in determining the probability that a continuous random variable falls within a particular range, providing a foundational aspect in probability theory and statistical analysis.
Poisson distribution: The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, under the condition that these events happen with a known constant mean rate and are independent of the time since the last event. This distribution is particularly useful for modeling rare events, such as the number of phone calls received at a call center in an hour or the number of accidents at a traffic intersection in a day.
Probability density function: A probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. The area under the PDF curve over an interval represents the probability that the variable falls within that interval.
Probability Density Function: A probability density function (PDF) is a statistical function that describes the likelihood of a continuous random variable taking on a particular value. The PDF provides the probabilities of the random variable falling within a particular range of values, which can be determined by calculating the area under the curve of the function within that range. This concept is crucial in understanding distributions such as binomial and normal distributions, where it helps illustrate how probabilities are distributed across different outcomes.
Probability Mass Function: A probability mass function (PMF) is a function that gives the probability of each possible value of a discrete random variable. It provides a complete description of the distribution of probabilities for all outcomes, ensuring that the total probability sums up to one. The PMF is essential for calculating probabilities related to discrete distributions, like the binomial distribution, which deals with the number of successes in a fixed number of independent Bernoulli trials.
Random variable: A random variable is a numerical outcome of a random process, serving as a way to quantify uncertainty. It can take on different values based on the outcomes of a particular experiment or event, and is often used to model real-world scenarios. Random variables can be classified into two types: discrete and continuous, and they are foundational to understanding probability distributions and statistical measures.
Success probability: Success probability is the likelihood of a particular outcome occurring in a binomial experiment, typically represented as 'p'. It reflects the chance of success in each trial and is a crucial component in calculating the binomial distribution, which models the number of successes in a fixed number of independent trials.
Trials: Trials are individual instances or repetitions of an experiment or process in probability. Each trial results in one of the possible outcomes.
Variance: Variance is a statistical measure that represents the degree of spread or dispersion of a set of values around their mean. It helps quantify how much the values in a data set deviate from the average, providing insight into the consistency and variability of the data. Understanding variance is essential in probability, distributions, and regression analysis as it influences predictions and expectations derived from data.
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7.11 Expected Value