Basics of Probability and Statistics Week 1 Challenge

Q1.

The formula for calculating the variance of a population and the variance of a sample are different because they are based on different sets of data.

The variance is a measure of how spread out a set of data is, and it is calculated by finding the average of the squared differences between each data point and the mean of the data set.

The formula for the variance of a population is: σ² = Σ(x - μ)² / N

where σ² is the population variance, Σ is the sum of the squared differences, x is each individual value in the population, μ is the population mean, and N is the total number of values in the population.

The formula for the variance of a sample is: s² = Σ(x - x̄)² / (n-1)

where s² is the sample variance, Σ is the sum of the squared differences, x is each individual value in the sample, x̄ is the sample mean, and n is the sample size.

The key difference between the two formulas is that the sample variance formula divides by (n-1) instead of n in the population variance formula. This is known as Bessel's correction, and it is used to correct for the fact that the sample mean is being used to estimate the population mean, which introduces some degree of uncertainty in the calculation.

In other words, the formula for the sample variance is adjusted to account for the fact that the sample is only a subset of the population and may not perfectly represent the population. By dividing by (n-1), the sample variance formula provides a slightly higher estimate of the true variance of the population, which is a more accurate reflection of the variability in the population based on the available sample data.

Q2.

In machine learning, stratified sampling and cluster sampling are two different techniques used for selecting a representative subset of data from a larger dataset.

Stratified sampling is a technique where the population is divided into several subgroups or strata based on a specific characteristic, and samples are selected from each stratum based on a proportionate random sampling strategy. This is often used when the population has distinct subgroups with different characteristics and it is important to ensure that each subgroup is represented in the sample to avoid biases. In stratified sampling, the goal is to obtain a representative sample from each subgroup, and the final sample is a combination of the samples obtained from each stratum.

On the other hand, cluster sampling is a technique where the population is divided into several clusters or groups, and a random sample of clusters is selected for analysis. Then, all of the elements in the selected clusters are included in the sample. This technique is often used when it is difficult or expensive to obtain a complete list of individuals in the population, but it is possible to identify clusters that are representative of the population. In cluster sampling, the goal is to obtain a representative sample of clusters, and all of the elements within the selected clusters are included in the sample.

In machine learning, these techniques are often used in the context of selecting a representative subset of data for training and testing a model. Stratified sampling can be used to ensure that each subgroup is represented in the training and testing data, which can help prevent biases in the model. Cluster sampling can be used when the dataset is very large and it is computationally expensive to analyze all of the data, and it is possible to identify clusters that are representative of the population. By selecting a random sample of clusters and analyzing all of the elements within those clusters, cluster sampling can provide a representative sample of the population with less computational resources.

Q3.

In machine learning, the number of different samples that can be created out of a population of size n depends on the sampling technique used and the size of the sample to be drawn.

Assuming a simple random sampling technique, where each element in the population has an equal chance of being selected, the number of different samples that can be created of size k can be calculated using the following formula:

n! / (k!(n-k)!)

where n is the population size, and k is the sample size.

For example, if the population size is 10 and a sample of size 5 is to be drawn, the number of different samples that can be created would be:

10! / (5!(10-5)!) = 252

Therefore, there are 252 different samples of size 5 that can be drawn from a population of size 10 using simple random sampling.

It's important to note that this formula assumes sampling with replacement, meaning that an element can be selected multiple times in a single sample. If sampling is done without replacement, meaning that an element can only be selected once in a sample, the number of different samples that can be created would be reduced.

Q4.

The probability of drawing 2 jacks from a standard deck of 52 cards can be calculated using the formula for calculating the probability of independent events:

P(A and B) = P(A) x P(B)

where P(A and B) is the probability of both events A and B occurring, and P(A) and P(B) are the probabilities of each individual event occurring.

First, we need to calculate the probability of drawing a jack on the first draw. There are 4 jacks in a deck of 52 cards, so the probability of drawing a jack on the first draw is:

P(jack on first draw) = 4/52 = 1/13

Assuming that the first card is not replaced, there are now 51 cards left in the deck, including 3 jacks. Therefore, the probability of drawing a jack on the second draw, given that a jack was not drawn on the first draw, is:

P(jack on second draw | no jack on first draw) = 3/51

Now, we can use the formula for the probability of independent events to calculate the probability of drawing 2 jacks from the deck:

P(2 jacks) = P(jack on first draw) x P(jack on second draw | no jack on first draw)

P(2 jacks) = (1/13) x (3/51)

P(2 jacks) = 0.0045 or approximately 0.45%

Therefore, the probability of drawing 2 jacks from a standard deck of 52 cards is approximately 0.45%.

Q5.

The probability of rolling two dice and getting the same number on both dice can be calculated by considering all the possible outcomes and the number of outcomes that result in both dice showing the same number.

There are 6 possible outcomes for each roll of a single die, and there are 6 x 6 = 36 possible outcomes when rolling two dice together. Of these outcomes, there are 6 outcomes where both dice show the same number: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).

Therefore, the probability of rolling two dice and getting the same number on both dice is:

P(same number on both dice) = number of outcomes where both dice show the same number / total number of possible outcomes

P(same number on both dice) = 6 / 36

P(same number on both dice) = 1 / 6

Therefore, the probability of rolling two dice and getting the same number on both dice is 1/6 or approximately 0.1667 (16.67%).