Binomial Distribution in very simple words

The binomial distribution is a probability distribution that describes the outcome of a series of independent "yes/no" experiments, or Bernoulli trials, in which there are only two possible outcomes. It is used to model the probability of a specific number of successes in a given number of trials, where the probability of success is the same for each trial.

For example, if you are flipping a coin and want to know the probability of getting a certain number of heads in a certain number of flips, you can use the binomial distribution to model this. If the probability of getting a heads on each flip is 0.5, you can use the binomial distribution to find the probability of getting, for example, 3 heads out of 10 flips.

The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success on each trial (p). The probability of a specific number of successes (x) in a given number of trials can be calculated using the following formula:

probability = (n choose x) * p^x * (1-p)^(n-x)

where "n choose x" represents the binomial coefficient, which is a way of selecting a specific number of items from a larger group without replacement.

The binomial distribution can be useful for modeling and analyzing a wide range of real-world situations, such as the probability of winning a game of chance, the probability of a medical treatment being effective, or the probability of a machine failing. It is a widely used and important concept in statistical analysis and probability.

Example:

Here is a simple example of using the binomial distribution to model the probability of a specific number of successes in a given number of trials:

Suppose you are flipping a coin and want to know the probability of getting exactly 3 heads in 10 flips. The probability of getting heads on each flip is 0.5, so the number of trials (n) is 10 and the probability of success (p) is 0.5. Using the formula for the binomial distribution, we can calculate the probability of getting 3 heads in 10 flips as follows:

probability = (10 choose 3) * (0.5^3) * (0.5^7)

= (1098)/(321) * (0.125) * (0.0078125)

= 0.1171875

So the probability of getting exactly 3 heads in 10 flips is approximately 0.12, or 12%.

This is just a simple example to illustrate how the binomial distribution can be used to calculate the probability of a specific number of successes in a given number of trials. In practice, the binomial distribution can be used to model and analyze a wide range of real-world situations.

Percentiles in simple words

A percentile is a value on a scale that indicates the percentage of a distribution that is equal to or below it. For example, if a value is at the 75th percentile, it means that 75% of the values in the distribution are equal to or below it.

To calculate the percentile of a value, you can use the following steps:

Arrange the values in the distribution in ascending order.

Find the position of the value you want to find the percentile for.

Calculate the percentile using the following formula: Percentile = (position of the value / total number of values) x 100

For example, let's say you have the following distribution of values: 3, 5, 7, 8, 9, 10, 12

If you want to find the percentile for the value 8, you would follow these steps:

Arrange the values in ascending order: 3, 5, 7, 8, 9, 10, 12

Find the position of the value 8: It is the 4th value in the list.

Calculate the percentile: (4 / 7) x 100 = 57.14%

This means that the value 8 is at the 57.14th percentile of the distribution.

Percentiles are useful for understanding how a value compares to the rest of the distribution. For example, if a student scored in the 90th percentile on a test, it means that they scored higher than 90% of the other students who took the test.

Standardizing Z scores simple explanation

Standardizing a value, or a z-score, is a way of expressing how many standard deviations a value is from the mean of a distribution.

To standardize a value, you can use the following formula:

z = (x - μ) / σ

Where x is the value you want to standardize, μ is the mean of the distribution, and σ is the standard deviation of the distribution.

For example, let's say you have a distribution with a mean of 100 and a standard deviation of 10. If you want to standardize the value 110, you would do the following calculation:

z = (110 - 100) / 10 = 1

This means that the value 110 is 1 standard deviation above the mean of 100.

Standardizing values can be useful in comparing values from different distributions or in identifying unusual values that fall outside of the normal range.

A more simple example 

To standardize your age of 10 years, you would need to know the mean and standard deviation of the age distribution you are comparing it to. For example, if you are comparing your age to the age of students in your class and the mean age of the students is 10 years and the standard deviation is 2 years, you can standardize your age as follows:

z = (10 - 10) / 2 = 0

This means that your age of 10 years is exactly the average age of the students in your class. If the mean age of the students was 9 years and the standard deviation was still 2 years, your standardized age would be:

z = (10 - 9) / 2 = 0.5

This means that your age is 0.5 standard deviations above the mean age of the students in your class.

It's important to note that standardizing a value only makes sense if you are comparing it to a distribution with a known mean and standard deviation. Without this information, you cannot accurately standardize a value.

Posterior probability

In probability theory, the posterior probability is the probability of an event occurring after taking into account new evidence or information. It is calculated using Bayes' theorem, which states that the posterior probability is equal to the prior probability (the probability of the event occurring before taking into account the new information) multiplied by the likelihood (the probability of observing the new evidence given that the event has occurred) divided by the marginal probability (the probability of observing the new evidence).

For example, let's say you have a box with 10 marbles in it, 5 of which are red and 5 of which are blue. You draw a marble from the box, observe that it is red, and then put it back in the box. The prior probability that the next marble you draw will be red is 5/10, or 50%. Now, let's say you draw another marble and observe that it is also red. The likelihood of observing this new evidence (a red marble) given that the event (drawing a red marble) has occurred is 1, because if you have already drawn a red marble, the probability of drawing another red marble is 1. The marginal probability, in this case, is the probability of drawing two red marbles in a row, which is (5/10) * (5/10), or 25%. Using Bayes' theorem, we can calculate the posterior probability of drawing a red marble as follows:

Posterior probability = Prior probability * Likelihood / Marginal probability

Posterior probability = (5/10) * 1 / (5/10) * (5/10)

= 1 / (5/10)

= 2/5

= 40%

So, the posterior probability of drawing a red marble after observing two red marbles in a row is 40%. This probability takes into account the new evidence (observing two red marbles in a row) and updates the prior probability (the probability of drawing a red marble before observing any evidence) accordingly.

Bayesian tree in simple words with an example

A Bayesian tree is a graphical representation of a Bayesian network, which is a type of probabilistic model used to represent the relationships between different variables and their probabilities.

A Bayesian tree is made up of nodes, which represent variables or events, and branches, which represent the relationships between the nodes. Each node in a Bayesian tree has a probability associated with it, which represents the likelihood of that event occurring.

Here's an example of a Bayesian tree:

Imagine you are trying to predict the likelihood of it raining tomorrow. You know that the probability of it raining depends on the weather forecast and the likelihood of the forecast being accurate. You can create a Bayesian tree to represent the relationship between these variables:

Rain (A)

/

Forecast (B) Accuracy (C)

In this example, the node "Rain" represents the event of it raining tomorrow. The nodes "Forecast" and "Accuracy" represent the variables that influence the probability of it raining. The branches connecting the nodes represent the relationships between the variables.

To calculate the probability of it raining tomorrow, you would use Bayes' theorem to combine the probabilities of the "Forecast" and "Accuracy" nodes. For example, if the probability of the forecast being correct is 0.9 and the probability of it raining given that the forecast is correct is 0.7, you can use Bayes' theorem to calculate the probability of it raining tomorrow:

P(A|B) = (P(B|A) * P(A)) / P(B)

P(Rain|Forecast) = (P(Forecast|Rain) * P(Rain)) / P(Forecast)

P(Rain|Forecast) = (0.7 * 0.5) / 0.9

P(Rain|Forecast) = 0.39

This example shows how a Bayesian tree can be used to represent the relationships between variables and their probabilities, and how Bayes' theorem can be used to make predictions or estimates about the likelihood of an event occurring.

Bayes' theorem

 Bayes' theorem is a mathematical formula that describes the relationship between the probability of an event occurring and the likelihood of certain evidence being present. It allows us to make predictions or estimates about the probability of an event occurring, based on past data or evidence.

Here's the formula for Bayes' theorem:

P(A|B) = (P(B|A) * P(A)) / P(B)

Where:

P(A|B) is the probability of event A occurring, given that event B has occurred.

P(B|A) is the probability of event B occurring, given that event A has occurred.

P(A) is the probability of event A occurring.

P(B) is the probability of event B occurring.

Bayes' theorem can be used in a variety of contexts, including decision-making, risk assessment, and statistical analysis. It is a widely used and important tool in many fields, including statistics, machine learning, and data analysis.

Example:

Let's say you have a box with some marbles in it. You know that some of the marbles are red and some are blue. You can use Bayes' theorem to figure out the probability that a marble you pull out of the box will be red, based on how many red and blue marbles you have seen before.

Imagine you pull out 5 marbles from the box, and 3 of them are red and 2 of them are blue. Using Bayes' theorem, you can calculate the probability that the next marble you pull out will be red.

First, you need to know the probability that a marble is red, given that it is red. This is called P(A|B), where A is the event of a marble being red and B is the evidence that the marble is red. In this case, the probability of a marble being red, given that it is red, is 1.0, because 100% of red marbles are red.

Next, you need to know the probability that a marble is red, given that it is not red. This is called P(B|A), where B is the event of a marble being red and A is the evidence that the marble is not red. In this case, the probability of a marble being red, given that it is not red, is 0.0, because 0% of non-red marbles are red.

Then, you need to know the probability of a marble being red overall. This is called P(A), and it is calculated by taking the total number of red marbles and dividing it by the total number of marbles. In this case, the probability of a marble being red is 3/5, or 0.6.

Finally, you need to know the probability of a marble being not red. This is called P(B), and it is calculated by taking the total number of non-red marbles and dividing it by the total number of marbles. In this case, the probability of a marble being not red is 2/5, or 0.4.

Now, you can plug all of these values into Bayes' theorem to calculate the probability that the next marble you pull out will be red:

P(A|B) = (P(B|A) * P(A)) / P(B)

P(A|B) = (1.0 * 0.6) / 0.4

P(A|B) = 1.5

The probability that the next marble you pull out will be red is 1.5, which is greater than 1.0. This means that, based on the data you have collected so far, it is more likely that the next marble you pull out will be red.

Conditional probability

In probability, the conditional probability of an event is the probability that the event will occur, given that another event has already occurred.

For example, let's say we have a deck of cards and we want to know the probability of drawing an Ace of Spades, given that we have already drawn the Ace of Hearts. In this case, the probability of drawing the Ace of Spades is 1 out of 51, since there is only 1 Ace of Spades left in the deck and 51 total cards remaining.

We can express this probability using the following formula:

P(Ace of Spades | Ace of Hearts) = (Number of ways to get the Ace of Spades given that the Ace of Hearts has already been drawn) / (Total number of cards remaining)

So in this case, the probability of drawing the Ace of Spades given that the Ace of Hearts has already been drawn is 1/51.

It's important to note that the conditional probability of an event is not the same as the probability of the event occurring on its own. The probability of an event occurring on its own is called the unconditional probability.

Another expample (from the lesson)

Last semester, out of 170 students taking a particular statistics class, 71 students were “majoring” in social sciences and 53 students were majoring in pre-medical studies. There were 6 students who were majoring in both pre-medical studies and social sciences. What is the probability that a randomly chosen student is majoring in social sciences, given that s/he is majoring in pre-medical studies?

1 point

(71+53−6)/170

6/170

6/71

6/53

To solve this problem, we can use the formula for conditional probability, which is the probability of an event occurring given that another event has already occurred. The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the probability of event A occurring given that event B has already occurred, P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.

In this problem, we are asked to find the probability that a student is majoring in social sciences given that they are majoring in pre-medical studies, so we can define event A as "majoring in social sciences" and event B as "majoring in pre-medical studies." We are given that there are 6 students who are majoring in both pre-medical studies and social sciences, so the probability of both events occurring is 6/170. We are also given that there are 53 students majoring in pre-medical studies, so the probability of event B occurring is 53/170. Plugging these values into the formula for conditional probability, we get:

P(A|B) = P(A and B) / P(B)

= (6/170) / (53/170)

= 6/53

So, the probability that a student is majoring in social sciences given that they are majoring in pre-medical studies is 6/53. The answer is therefore option 4: 6/53.