**ST112 Probability Assignment Sample NUIG Ireland**

Probability is a mathematical tool that can be used to calculate the chances that an event will occur. In the context of ST112, probability can be used to determine the likelihood that a particular sequence of nucleotide bases will form during replication.

Many different factors contribute to the probability of a particular sequence forming, including the base composition of the DNA molecule, the temperature of the environment, and the relative concentrations of free nucleotides. By understanding how these factors interact, scientists can develop models that predict how likely it is for a particular sequence to form under specific conditions.

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In this section, we are describing some assigned tasks. These are:

**Assignment Task 1: Demonstrate the concepts of systematic and random variation, and that probability is concerned with the construction of mathematical models for random phenomena that are subject to stable relative frequencies; comprehend that probability and (inferential) statistics are opposite scientific processes, and be able to give examples where the former is used to justify statistical inferences made in the real world.**

Probability is concerned with the construction of mathematical models for random phenomena. A model is a simplification of a complex reality. In statistics, a model is usually expressed in terms of probability distributions. A probability distribution is a function that assigns probabilities to all the possible outcomes of some random event. The most famous probability distribution is the Gaussian or normal distribution, which arises when events are the result of repeated trials with independent results.

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Systematic variation occurs when there is a consistent pattern in the results of repeated measurements. For example, if you repeatedly measure the height of people in your class, you will find that they vary in height, but there will be some people who are consistently taller than others. This variation is due to systematic factors, such as genetics.

Random variation, on the other hand, is due to chance. If you measure the height of people in your class again, you will find that the heights vary from person to person, but there is no consistent pattern. This variation is due to random factors, such as the fact that people are of different heights.

**Assignment Task 2: Demonstrate the role of probability both as a discipline in its own right with applications to e.g. financial decision-making, gambling, communications systems), and as the tool used in justifying statistical inferences (i.e. in justifying statements made about entire populations based on information available in samples taken from the populations).**

Probability is a mathematical discipline that quantifies the likelihood of events. Probability theory is used to model uncertain situations and to make predictions about the outcomes of events. Probability theory is also used in statistics, machine learning, artificial intelligence, and computer science.

Probability theory has applications in many filed of study, including finance, insurance, biology decision-making, gambling, gaming, lottery, market research medical analysis communication systems, and more. In finance, the probability is used to price options and calculate risks associated with financial investments. Insurance companies use probability theory to calculate premiums and policy limits. Decision-makers often use probabilities to quantify the risks associated with making decisions. Gamblers use probability theory to calculate the odds of winning and to make bets. Market researchers use probability theory to design surveys and to make predictions about the results of marketing campaigns.

Statistical inference is the process of making inferences about a population based on information from a sample. In other words, statistical inference is the process of using data from a sample to make claims about a population. The goal of statistical inference is to estimate population parameters, such as the mean, median, and standard deviation. Statistical inference is used in many fields, including sociology, psychology, epidemiology, biology, economics, and finance.

There are two types of statistical inferences: point estimation and hypothesis testing. Point estimation is the process of estimating a population parameter using a single sample statistic. For example, the sample mean is a point estimate of the population mean. Hypothesis testing is the process of using data from a sample to test a hypothesis about a population.

Statistical inference is usually divided into two types: inductive and deductive. Inductive statistical inference is based on the idea that the data in the sample can be used to make predictions about the population. In other words, the inductive inference is based on the idea that the data in the sample can be used to make claims about a population. Deductive statistical inference is based on the idea that the data in the sample can be used to test hypotheses about a population. In other words, deductive inference is based on the idea that the data in the sample can be used to test claims about a population.

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**Assignment Task 3: Demonstrate the frequentist and classical approaches to probability, be able to calculate probabilities for compound events, understand the ideas of mutually exclusive events and independent events, and be able to perform calculations involving Bayes’ formula.**

The frequentist approach to probability is the most commonly used in scientific experiments. It assigns a probability to an event based on how often that event occurs in a large number of trials.

The classical approach to probability is sometimes used in real-world situations where we can’t do lots of trials. It assigns a probability to an event based on our knowledge of all the possible outcomes and their probabilities.

We can calculate the probability of compound events by multiplying together the individual probabilities for each outcome. For example, if we wanted to know the probability of getting heads and tails when flipping a coin twice, we would multiply together the two individual probabilities (1/2 x 1/2 = 1/4).

Mutually exclusive events are events that cannot happen at the same time. For example, flipping a coin can result in either heads or tails, but not both.

Independent events are events that are not affected by each other. For example, flipping a coin does not affect whether or not it will rain tomorrow.

Bayes’ formula is a way of calculating the probability of an event happening, given that another event has already happened. For example, we might use Bayes’ formula to calculate the probability of a person having a disease, given that they have tested positive for the disease.

**Assignment Task 4: Demonstrate the motivation for the introduction of the concept of a random variable and the idea that a given population can be viewed as synonymous with the distribution of a suitably-defined random variable.**

The concept of a random variable was introduced to provide a mathematical model for understanding and analyzing concepts related to probability and statistics. A random variable is defined as a function that assigns a numerical value to each element in a sample space. The idea behind this is that if we can identify and quantify the range of possible values that a random variable can take on, then we can begin to understand the distribution of probabilities associated with that variable.

There are two main types of random variables- discrete and continuous. Discrete random variables take on values that are isolated and distinct from one another, such as integers or whole numbers. On the other hand, continuous random variables can take on any value within a certain range, such as minutes, seconds, or even hours.

The distribution of a random variable is a mathematical function that describes how the values of the random variable are distributed. The most commonly used distribution is the normal (or Gaussian) distribution, which is defined by a bell-shaped curve. Other less commonly used distributions include the uniform distribution and the exponential distribution.

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**Assignment Task 5: Model basic discrete random variables and perform calculations based on hypergeometric, multivariate hypergeometric, binomial, geometric, negative binomial, and Poisson distributions.**

Discrete random variables can be classified into several categories, depending on the type of distribution they follow. The most common types of discrete distributions are hypergeometric, multivariate hypergeometric, binomial, geometric, and negative binomial distributions.

The hypergeometric distribution is used when dealing with situations in which a population is divided into two or more groups, and a sample is drawn from one of these groups. The multivariate hypergeometric distribution is similar to the hypergeometric distribution but takes into account that there may be more than one group being sampled.

The binomial distribution applies when an experiment consists of a series of independent trials, each of which has only two possible outcomes: success or failure. The geometric distribution is used when an experiment has a single success or failure outcome and the probability of success is the same for each trial.

The negative binomial distribution is used when there are a fixed number of failures in a series of independent trials, and the probability of success is the same for each trial. The Poisson distribution is used to model the number of events occurring in a given period, such as the number of car accidents in a given year.

**Assignment Task 6: Demonstrate the importance of the first two moments of discrete and continuous random variables as summary measures of distribution, and be able to compute the mean and variance of certain discrete variables.**

The first two moments of a random variable are the mean and the variance. The mean is a measure of the central tendency of the distribution, and the variance is a measure of the dispersion of the distribution.

The mean of a discrete random variable is simply the sum of all the possible values that the variable can take on, multiplied by their respective probabilities. The variance of a discrete random variable is the sum of the squares of the differences between each possible value and the mean, multiplied by their respective probabilities.

The mean of a continuous random variable is the integral of the variable’s probability density function (PDF) over all possible values. The variance of a continuous random variable is the integral of the square of the difference between the variable’s PDF and the mean, over all possible values.

The mean and variance are important summary measures of distribution because they provide information about the centre and spread of the distribution. They can also be used to calculate other important measures of dispersion, such as the standard deviation and the coefficient of variation.

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**Assignment Task 7: Demonstrate the idea underlying the density of a continuous random variable and be able to perform probability calculations for normally distributed variables.**

The density of a continuous random variable is a measure of the probability that the value of the random variable falls within a certain range. The higher the density, the more likely it is that the random variable will take on a particular value.

To calculate the density of a normal distribution, you need to know two things: the mean and standard deviation. The mean is simply the average value of the distribution, and the standard deviation is a measure of how spread out the values are. Once you have those two values, you can use this formula to calculate the density:

density = (1 / (2 * pi)) * sqrt( (mean) ^ 2 / (standard deviation) ^ 2 )

This formula tells you that the density of a normal distribution is determined by its mean and standard deviation. The higher the mean, the more dense the distribution will be. The lower the standard deviation, the more dense the distribution will be.

You can use this formula to calculate the probability that a random variable will take on a particular value. For example, if you know that the mean of a normal distribution is 5 and the standard deviation is 2, you can calculate the probability that the random variable will be between 4 and 6 as follows:

P(4 < X < 6) = (1 / (2 * pi)) * sqrt( (5 – 4) ^ 2 / (2) ^ 2 ) – (1 / (2 * pi)) * sqrt( (6 – 5) ^ 2 / (2) ^ 2 )

P(4 < X < 6) = 0.39894

This means that there is a 39.9% chance that the random variable will be between 4 and 6.

You can also use the density to calculate the probability that a random variable will be less than or equal to a particular value. For example, if you know that the mean of a normal distribution is 5 and the standard deviation is 2, you can calculate the probability that the random variable will be less than or equal to 4 as follows:

P(X <= 4) = (1 / (2 * pi)) * sqrt( (5 – 4) ^ 2 / (2) ^ 2 )

P(X <= 4) = 0.25

This means that there is a 25% chance that the random variable will be less than or equal to 4.

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**Assignment Task 8: Demonstrate the importance and properties of sampling distributions, especially that of the sample mean; be able to calculate probabilities about the mean of a random sample when sampling from a normal distribution.**

The importance of sampling distributions cannot be understated. They provide critical information about the distribution of a given variable and allow us to make inferences about that variable. In particular, the sampling distribution of the sample means is very important, as it allows us to calculate probabilities about the population means.

There are several key properties of sampling distributions that are important to note. First, they are typically bell-shaped, which makes them easy to work with mathematically. Second, they are generally symmetric around the mean. Third, they have a defined variance and standard deviation. fourth, they are relatively easy to work with computationally.

All of these properties make sampling distributions an essential tool for statisticians and researchers alike.

When sampling from a normal distribution, the mean of the sample will be normally distributed as well. This means that we can use the properties of the normal distribution to calculate probabilities about the population mean. For example, if we know that the mean of a population is 10 and the standard deviation is 2, we can calculate the probability that a random sample of size 100 will have a mean between 9 and 11 as follows:

P(9 < Xbar < 11) = (1 / (2 * pi)) * sqrt( (10 – 9) ^ 2 / (2) ^ 2 ) – (1 / (2 * pi)) * sqrt( (11 – 10) ^ 2 / (2) ^ 2 )

P(9 < Xbar < 11) = 0.39894

This means that there is a 39.9% chance that the mean of a random sample of size 100 will be between 9 and 11.

**Assignment Task 9: State the central Limit Theorem and apply it to compute probabilities relating to sums and means of values of both quantitative and Bernoulli variables.**

The central limit theorem states that, under certain conditions, the sum of a large number of independent random variables will tend to be normally distributed. This means that if you take the mean of a large sample of independent random variables, the result will tend to be normally distributed.

You can use the central limit theorem to compute probabilities relating to sums and means of quantitative and Bernoulli variables. For example, if you have a large sample of independent Bernoulli variables with success probability p, then the mean of those variables will tend to be distributed according to a normal distribution with mean p and standard deviation √p(1-p). This means that you can use the central limit theorem to compute probabilities such as “What is the probability that the mean of my sample is greater than 0.5?”

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