The post Probability appeared first on A Level Maths.

]]>- Two events that have nothing in common are known as mutually exclusive events
- Two events that have elements in common are known as non-mutually exclusive events
- If two events are independent event meaning they both do not depend on one another
- If two events are dependent events they both are (conditional probability)

It is a measure of the likelihood that an event will happen. Meaning, imagine that you roll a dice the possible outcomes of rolling a dice would be {1, 2, 3, 4, 5, 6} that is a set of outcomes, which describe what outcomes correspond to the “event” happening. For example, “rolling a multiple of 3” corresponds to the set of outcomes {3, 6}. We can thus say that the probability of rolling a multiple of 3 is the favourable outcomes divided by the possible outcomes.

So in this case we can say that the favourable outcome is {3, 6} since that’s the event we are looking for. however, the possible outcomes would be {1, 2, 3, 4, 5, 6} which would be the total outcomes we can get.

Similarly if we toss a coin twice what is the likelihood that we get tails both time.

By using the formula above we can say that the favourable outcome in this case would be:

1 = TT

But the possible outcome would be:

4 = TT, TH, HT, HH

Thus, the probability would be .

We can also study probability using the set notations. A set is a collection of different elements.

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

**Empty set**

An empty set is an imaginary set which has no elements

= { } if there are no elements it is known as an empty set.

Note: is called Phi

**Subset**

Is the set in which all the elements are taken from a bigger set.

E.g

A = {2, 4, 7}

”A” is a subset of S as all the elements in ”A” are also present in S.

It can be written as:

**Improper and proper subset**

Every set has two improper subsets:

- An empty set
- And the set itself

E.g

S = {1, 2, 3, 4}

C = { }

B = {1, 2, 3, 4}

C and B are both subsets of S, C is a null set and B is the set itself

**Proper subset**

A proper subset is a set which has only a few elements of the main set.

E.g

A = {1, 2, 3, 4}

”A” here is the main subset and both B and C have few elements of A in their set.they are thus, known as a propers subset of A.

B = {1}

C = {1, 2}

All possible outcomes of a random experiment

E.g

S = {1, 2, 3, 4, 5, 6} when you roll a dice

S = {H, T} when you toss a coin

S = {HH, HT, TH, TT} when you toss 2 coins

In order to find A’ (A compliment) we subtract set A from the universal set to get A’ and the same is done with set B to find B complement:

A’ = U – A

B’ = U – B

*NOTE:*

P(A) + P(A’) = 1

**Events**

Any subset of a sample space is known as an event.

**Mutually Exclusive Events**

Two events that have nothing in common are known as mutually exclusive events or disjoint event. In other words if the intersection of two events is none then it’s mutually exclusive.

E.g

S = {2, 3, 4, 5, 6}

A = {1, 5}

B = {3, 6}

The venn diagram below shows mutually exclusive events.

However, keep in mind that B alone is not the complement of A.

Complement of A = {2, 3, 4, 6}

So we can write this as:

{ }

i.e (known as the addition law of probability for mutually exclusive events)

**Non – Mutually Exclusive Events**

If both events have something in common:

E.g

S = {1, 2, 3, 4, 5, 6}

C = {3, 4, 5}

D = {4, 5, 6}

Thus it can be written as:

{ }

(known as the addition law of probability for non-mutually exclusive events)

If two events are independent event meaning they both do not depend on one another.

A and B then the probability of A and B is equal to the probability A and probability B

E.g

If there are two sweets in a bag and 7 red coloured and 11 green coloured they are considered independent if the sweet drawn is replaced after pulling out, meaning put back inside the bag. In that case its tree diagram will look like the following:

However if the sweet drawn is not replaced, meaning it is not put back in the bag, in that case the tree diagram will look like the following:

Is the probability of one event, given that another event has already occured.

If two events are dependent events they both are:

If both the events are independent then we write it as:

E.g

Female | Male | Total | |
---|---|---|---|

Maths | 4 | 14 | 18 |

Economics | 17 | 41 | 58 |

Science | 4 | 25 | 29 |

Arts | 28 | 11 | 39 |

Total | 53 | 91 | 144 |

a) A student is selected at random from all of the students. Find the probability that

- Student is female

- Student is studying arts

b) the student is a female and studying arts

c) state whether F and A are independent

Since, they are not equal we can say that they depend upon each other.

d) A male student is selected find the probability that he studies Economics

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]]>The post The Uniform Distribution appeared first on A Level Maths.

]]>- In uniform distribution the random variable is a continuous random variable
- The probability density function is calculated as:
- Mean
- Variance
- The cumulative distribution function is calculated by integrating the probability density function f(x) to give
- Standard deviation is the under root of variance

In uniform distribution you should know that random variable is a continuous random variable. In continuous uniform distribution it takes infinite number of real values in an interval.

For example, if we say that it is observed in a school, over a period of 2 months that the teacher arrives in school earliest by 4 minutes, before school starts or latest by 6 minutes after school starts.

*(note: we are assuming that school starts at sharp 8 in the morning)*

Thus looking at the figure below, we can deduce that the probability of a teacher entering class anywhere between 7:56 to 8:06 is constant.

However, in order to find probability density function, we can say that, we know that the since, -4 till 6 are the only possibilities, we can thus say that the probability is greater than 0 only if the interval of x is between -4 and 6 .

Otherwise we can say that it is equal to 0 ( meaning, the teacher has taken a leave).

We can thus write the uniform distribution as:

X ~ U(A, B)

Which in this case will be:

X ~ U(-4, 6)

Now, in order to find the interval between A and B we use the probability density function. Which is calculated as:

In addition, if you are asked in a question, to find the mean we use the following formula:

Mean

But, we also need to understand how to derive this formula:

Variance

The cumulative distribution function of a continuous random variable, is known give the probabilities and is calculated by integrating the probability density function f(x) between the limits and x.

Q. A random variable is uniformly distributed over the interval 2 to 10. Find its variance.

*Solution:*

We have been given the interval 2 to 10 so we know, a = 2 and b = 10

The formula for variance is .

Plug, in the values to calculate the answer:

*Ans*

Q. A zero mean random signal is uniformly distributed between limits -a and +a and its mean square value is equal to its variance. Then the standard deviation of the signal is?

*Solution:*

We know that standard deviation is the under root of variance thus:

Next, the values of a and b are given to us:

a= -a and b = +a

Now plug the values into the formula above to get

**Ans**

Q. Suppose a train is delayed by approximately 60 minutes. What is the probability that train will reach by 57 minutes to 60 minutes?

*Solution:*

Interval of probability distribution a = 0 minutes b = 60 minutes

We need to find the probability density function with the given values of a and b between the probability :

We can thus, say the probability that train will reach by 57 minutes to 60 minutes is .

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]]>The post The Product Moment Correlation Coefficient appeared first on A Level Maths.

]]>- Formula for correlation
- r >0 positive correlation
- r <0 negative correlation
- r =0 no/zero correlation
- r =+1 r = -1 perfect positive and negative correlation respectively
- A good relation between the variables means that the line of best fit will pass through maximum points

The interdependence of the two variables is known as as correlation.correlation is measured by coefficient of correlation which is denoted by ”r”. And its numerical value ranges from +1 to -1.

It also helps us understand the strength of the relationship and whether the relationship between two variables is positive or negative. It’s formula is:

**1. Positive correlation:**

- Is when both the variables have the same type of moment and they both rise or fall together in the same direction
- E.g sale of ice cream with change in temperature, if the temperature increases more ice cream is sold and as the temperature decreases less ice cream is sold
- The expenditure of a family depends on their income. If income falls the expenditure also falls and vice versa.

**2. Negative correlation:**

- When both the variables move in different direction. They are considered to have an inverse relationship
- E.g when you start exercising your weight reduces significantly
- As the price of one product increases its demand falls and as its price decreases its demand increases.

**3. If the correlation is found to be 0** then in that case, both the variables X and Y are considered independent and are considered to have no linear dependency on each other.

- E.g The price of shoes and jeans have nothing in common, thus if the price of jeans increases or falls it will have no effect on the price of shoes and so we can say that they have zero correlation with each other

**4. If the coefficient of correlation is -1** it is considered a perfect negative correlation and **if the correlation is +1** then it is considered a perfect positive correlation. The closer the value is to -1 or +1 the stronger the relationship is considered to be.

Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

x | 9 | 11 | 12 | 13 | 15 | 18 | 16 | 14 | 12 | 10 |

y | 429 | 350 | 360 | 300 | 225 | 200 | 230 | 280 | 315 | 410 |

= 130

= 3090

= 38305

= 1760

= 1007750

a) Find the correlation coefficient.

We will first find the mean of x and y:

Next we will plug in all the values in the formula of correlation coefficient that we studied above,

b) Calculate the least square regression line.

Use the formula that we discussed in linear regression chapter to calculate this line.

Now plug in the values in the equation below and calculate ”a”.

*y = a+bx*

*309 = a -26.64 (13)*

*a = 655.56*

*y = 655.36 – 26.64 x*

c) Based on the regression line what will be the predicted loss from the company when there are 17 workers on duty?will you trust this value? Justify your answer.

Since the number of workers is an x variable, we will replace x with 17 the regression equation and calculate the loss to the company.

*y = 655.36 – 26.64(17)*

*y = 202.48*

Since the value of y is 202.48 which is not a an outlier. And the correlation coefficient is also close to -1 representing a strong negative correlation. Thus we can say that this value can be trusted.

- https://explorable.com/statistical-correlation
- J.S Abdey Statistics 1

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]]>The post The Poisson Distribution appeared first on A Level Maths.

]]>- The probability function of the poisson distribution is
- Both the mean and variance the same in poisson distribution.

When calculating poisson distribution the first thing that we have to keep in mind is the if the random variable is a discrete variable. If however, your variable is a continuous variable e.g it ranges from 1<x<2 then poisson distribution cannot be applied.

The possible values of the poisson distribution are the non-negative integers 0,1,2… The probability function of the poisson distribution is:

Where (pronounced lamda) is the mean, which is calculated as [n.p]

Where **n** is the total number of trials and

**P** is the successful probability

*Note: Always remember that both x and p will always stay associated with each other e.g if x is a success trial then p will also be a success trial.*

Both the mean and variance the same in poisson distribution

in this case is considered as the parameter of the distribution. We are already familiar with the idea that poisson distribution occurs when there are discrete events in a continuous time.

E.g we can say that if you have 4 phone calls in one hour, then the number of phone calls at any given time in that hour has poisson distribution with parameter 4.

The mean of the poisson distribution would be:

The variance of the poisson distribution would be:

Poisson distribution has many properties like:

- The trials are independent
- The events cannot occur simultaneously
- Events are random and unpredictable
- The poisson distribution provides an estimation for binomial distribution.

Consider the sum of two independent random variables X and Y with parameters L and M. Then the distribution of their sum would be written as:

Thus,

Q. A hospital board receives an average of 4 emergency calls in 10 minutes. What is the probability that there are at most 2 emergency calls?

*Solution:*

**Step #1**

We will first find the and x.

also known as the mean or average or expectation, has been provided in the question.

= 4

its less than equal to 2 since the question says at most. Which means, maximum 2 not more than that.

**Step #2**

We will now plug the values into the poisson distribution formula for:

P[ \le 2] = P(X=0) + P(X=1)+(PX=2)

The mean will remain same throughout, however, the value of x will change (0, 1, 2)

Note: the value of e will remain constant .

**Step#3**

Add all the answers together to get the final answer.

P[latex]\le \quad 2[/latex] = 0.0183 + 0.7328 +0.1465

= 0.8976

Q. Births in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability of observing 4 births in a given hour at the hospital?

**Step#1**

Calculate both and X

= 1.8

X = P(X = 4)

This question does not state anything else but to find the probability of 4 births in an hour, thus we will just simply calculate P(X = 4)

**Step#2**

We will now use the formula of poisson distribution to calculate the answer.

We can thus say, the probability of observing 4 births in a given hour at the hospital is 0.0723

- http://www.stats.ox.ac.uk/~marchini/teaching/L5/L5.notes.pdf

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]]>The post The Normal Distribution appeared first on A Level Maths.

]]>- The normal distribution has a bell shaped curve
- It is symmetric about the mean
- Each normal distribution is affected by the either the mean() or standard deviation()
- The total area under the normal curve is equal to 1
- To find the normal distribution we use the following formula:

The normal distribution is a theoretical distribution of values. You will already be familiar with its bell shaped curve(shown below).

The normal distribution has many characteristics such as its single peak, most of the data value occurs near the mean, thus a single peak is produced in the middle. Secondly, it is symmetric about the mean. That is, the distributions of values to the right and left of the mean are mirror images, which shows that the distribution, lastly, tapering. Meaning, the further you get from the mean the fewer data is represented thus, it can be said that the distribution tapers out in both direction.

The graph below shows the probability density function of normal distribution.

Normal distribution is affected by various factors. That is, whenever there are several different factors, affecting the outcome. We tend to produce a graph like normal distribution.

Moreover, each normal distribution is affected by the either the mean() or standard deviation().

.

In normal distribution, mean is also considered the median as it tells us the midpoint of the graph. As it is shown in the graph below, the line of symmetry.

In order to find the normal distribution we use the following formula:

The average on a statistics test was 78 with a standard deviation of 8. If the test results are normally distributed, find the probability that a student receives a test score less than 90.

**Step 1**

We know that average is also known as mean. So our mean is 78 and are standard deviation is 8. We also know that 90 is more than the mean, so it will be drawn to the right of the mean. We will now draw our normal distribution curve. And find the value of the shaded region.

**Step 2**

We will now, put both the values in the formula. To find the normal distribution of P(X < 90)

**Step 3**

We will check the value P(X < 90) = P(X < 1.5) from our z score table, under 1.5 and get the answer 0.9332.

We can thus conclude our answer by saying, the probability that a student receives a test score less than 90 is 0.9332.

The average height of 120 students of class 8 in a school is 150cm and the standard deviation is 10cm. Assuming that the height is normally distributed, determine how many students are more than 170cm and less than 12 cm.

**Step 1**

In this problem the data given to us is as follows:

mean = μ = 150

standard deviation = σ = 10

We will first draw our normal distribution graph. Just like we did in the previous example.

Since, we are looking for a value greater the 170 cm thus our graph will look like above, and we will subtract 1 from the final answer to get the value of greater than 170 cm.

**Step 2**

We will plug in the values in the formula.

The z score for 170cm would be:

**Step 3**

From the z score table we find that:

P(z ≥ 2)

= 1−0.977251

= 0.02275

Thus the number of students having height more than 170cm can be expected to be:

0.02275×120 = 2.73

≈ 3 students

b)

The z score for 125cm is:

From the normal distribution z score table we find that the P value for z = −2.5 is:

P(z ≤ −2.5) = 0.00621

Thus the number of students having height less than 125 cm would be:

0.00621 × 120 = 0.7452

≈1

So we cannot expect more than 1 student to have a height less than 125 cm.

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]]>The post The Geometric Distribution appeared first on A Level Maths.

]]>- The geometric distribution has a single parameter (p) = X ~ Geo(p)
- Geometric distribution can be written as ,

where*q = 1 – p* - The mean of the geometric distribution is:
- The variance of the geometric distribution is:
- The standard deviation of the geometric distribution is:

The geometric distribution are the trails needed to get the first success in repeated and independent binomial trial. Each trial has two possible outcomes, it can either be a success or a failure. We can write this as:

- P(Success) = p (probability of success known as p, stays constant from trial to trial).
- P(failure) = q (probability of failure is a complement of success, thus for any trial it remains
*1 – p*).

Hence, we can write its probability density function as:

In other words:

For x = 1, 2, 3….

The geometric distribution has a single parameter, the probability of success (p). We can thus write this as:

X ~ Geo(p)

Moreover, the mean and variance are the functions of p.

- The mean of the geometric distribution is:

- The variance of the geometric distribution is:

- The standard deviation of the geometric distribution is:

Q. In a large population of school students 30% have received karate training.

If students from this population are randomly selected, calculate:

a) what is the probability that the 6th person that was chosen at randomly was the first student to have received the karate training.

*Solution:*

Here we require the number of trials needed to get the first success. Since students are randomly selected from a large population, we can say that the trials are independent.

In addition, the fact that the first person has or has not received the karate training, does nothing to solve our query of whether the next randomly selected person has received the training or not.

To sum up what was said above:

- Independent trails have been repeated
- We want to know the number of trials required to get the first success
- The first student who got the karate training
- And lastly, the probability of success is constant

We can hence, say:

P = 0.3 (30%)

We want the probability of randomly selected 6 students thus we will write:

*Note: We want the first 5 students to not have received the karate training and the 6 to have.*

Thus, calculating the above equation we get:

(when rounded off)

We can thus conclude the answer by saying;

The probability that the 6th person that was chosen randomly was the first student to have received the karate training is 0.0504.

b) Find the mean, variance and the standard deviation of the example above.

*Solution:*

We will use the formulas required to find the three of them.

For mean:

For the variance:

Lastly, for the standard deviation:

c) What is the probability that the first student trained in karate occurs on or before the 3rd person was sampled.

*Solution:*

Now, we need to find the probability that the random variable X is less than equal to 3.

We will write this as:

Using, this formula, we can say:

= 0.657

In a nutshell, we can say that the probability that the first student trained in karate occurs on or before the 3rd person was sampled is 0.657.

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]]>The post The Discrete Uniform Distribution appeared first on A Level Maths.

]]>- The values of a discrete random variable are obtained by counting, thus making it known as countable
- Uniform distribution simply means that when all of the random variable occur with equal probability
- A random variable with probability density function is
- the expectation and variance of the data we use the following formulas

You must already be aware of the fact that there are two types of random variables, discrete random variables and continuous random variables. We have already discussed continuous random variable in this chapter we will discuss about the discrete random variable. The values of a discrete random variable are obtained by counting, thus making it known as countable.

You should by now also be aware of these two basic characteristics of a A discrete probability distribution function (PDF).

Each probability is between zero and one, inclusive.

The sum of all the probabilities is always equal to one.

Now that discrete random variable is clear. Lets, discuss uniform distribution.

Uniform distribution simply means that when all of the random variable occur with equal probability. Suppose that the random variable X can assume n different values. Also consider n is constant. Then we can say that:

For all values of = 0, 1, 2…. N

P(X = x) = 0 for other values of

Q. Suppose a fair die is rolled. What is the probability that the die will land on 5?

*Solution:*

We are already aware that a die has 6 sides, thus when a die is rolled we already know that the outcome will be one of the following {1, 2, 3, 4, 5, 6}.

We also know that each possible outcome is a random variable (X), and since the die is fair all the outcomes have equal chances to occur ().

Thus, we have a uniform distribution.

Therefore we can say:

In order to find the expectation and variance of the data we use the following formulas:

We can thus write it as .

Q. If we repeat the dice rolling experiment from above (Example #1) and we change the question a little bit from what is the probability that it will land on 5 to what is the probability that the die will land on a number that is smaller than 5?

*Solution:*

When already know that we have a uniform distribution as we discussed before that when a die is rolled, there are 6 possible outcomes that it can land on {1, 2, 3, 4, 5, 6}, with equal chances on landing on either number (if the die is fair of course).

The probability that the die will land on a number smaller than 5 is equal to:

P( X < 5 ) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

*Note: We did not take P(X < 5) because it’s not smaller than equal to 5. If the question said only then we would take P(X = 5).*

- https://revisionmaths.com/advanced-level-maths-revision/statistics/discrete-uniform-distribution
- https://www.texasgateway.org/resource/41-probability-distribution-function-pdf-discrete-random-variable
- https://stattrek.com/probability-distributions/discrete-continuous.aspx?Tutorial=stat

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]]>The post The Central Limit Theorem appeared first on A Level Maths.

]]>- The Central Limit Theorem (CLT) basically tells us that the sampling distribution of the sample mean is, at least approximately, normally distributed
- The Central Limit Theorem is exactly what the shape of the distribution of means will be when repeated samples from a given population are drawn
- As the sample size increases, the sampling distribution of the mean, X-bar, can be approximated by a normal distribution with mean µ and standard deviation

The Central Limit Theorem (CLT) basically tells us that the sampling distribution of the sample mean is, at least approximately, normally distributed. In fact, the central limit theorem applies regardless of whether the distribution of the is discrete (for example, Poisson or binomial) or continuous.

Now to define the central limit theorem we say that as the sample size increases, the sampling distribution of the mean, X-bar, can be approximated by a normal distribution with mean µ and standard deviation where:

µ is the population mean

σ is the population standard deviation

n is the sample size

Let’s study an example to be able to understand this better.

Q. Suppose the outcomes of a random phenomenon are normally distributed with a mean of μ=4.6 and a standard deviation of σ =0.8. Suppose a sample of size 12 is taken from this population and the mean of this sample, , is calculated.

(i) According to the Central Limit Theorem, what is the mean of the sampling distribution for ?

(ii) According to the Central Limit Theorem, what is the standard deviation of the sampling distribution for ?

*Solution:*

(i) As we know that the central limit theorem states that the mean of the sampling distribution for a statistic is the related population parameter value. In this example, the parameter value is the population mean. Thus, the mean of the sampling distribution is 4.6.

This implies that if we take several samples from the current given population and calculate the mean of each of these samples, the data set that consists of those means would be centered at the actual mean value for the entire population, hence the mean of the sampling distribution for is 4.6.

(ii) We know that the standard deviation for the sampling distribution is given by

So if we take samples of size 12 n=12, we get a standard deviation for our sampling distribution of:

**Ans**

We must note that the standard deviation for the sampling distribution is smaller than the standard deviation for the original parent population.

Q. An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size n = 25 are drawn randomly from the population.

(i) Find the probability that the sample mean is between 85 and 92.

*Solution:*

Let X = one value from the original unknown population.

Since the question asks you to find a probability for the sample mean (or average).

Let X = the mean or average of a sample of size 25.

Since ,

, and n = 25;

then

The probability that the sample mean is between 85 and 92 is 0.6997

P (85 < X < 92) = 0.6997

- http://www2.nau.edu/mat114-c/ch2g.php
- https://www.qualitydigest.com/inside/twitter-ed/understanding-central-limit-theorem.html
- http://www.stat.wmich.edu/s160/book/node43.html
- https://newonlinecourses.science.psu.edu/stat414/node/177/
- https://www.webassign.net/question_assets/idcollabstat2/Chapter7.pdf

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]]>The post The Binomial Distribution appeared first on A Level Maths.

]]>- The formula for binomial distribution is as follows:
- We write the binomial distribution as X ~ Bin(n, p)
- E(X) = np
- variance(X) = npq
- Standard deviation =

Binomial distribution is a discrete probability distribution. It has four major conditions that we need to keep in mind when dealing with binomial distribution.

There are fixed number of trials in a distribution, known as *n*.

Each event is an independent event, and the probability of each event is a mutually exclusive event.

Each trial has two outcomes e.g success or failure, heads or tails and pass or fail.

The probability of success for each trial will remain the same. Meaning, total probability of both events is equal to 100% e.g if the probability that there are 0.2 (20%) females students in a class, thus is automatically means that the probability of male students in a class 0.8(80%).

The formula for binomial distribution is as follows:

Where:

n= number of trials {0, 1, 2, 3….. n}

x = number of success

p = probability of success

q = probability of failure (1- p)

We write the binomial distribution in the form:

X ~ Bin(n, p)

Where:

X ~ Bin means X has a binomial distribution

n = total number of trials, which can be any number greater than 0

p= the probability of success, which can be any number between 0 and 1

We can thus say that the mean the variance and the standard deviation of the binomial distribution can be calculated by using the following formulas.

E(X) = np

variance(X) = npq

Standard deviation =

Q. If a fair coin is tossed, 20 times what is the probability of getting 5 heads.

*Solution:*

As the question says fair coin, we need to keep in mind that there are 50% chances of heads and 50% chances of tails.

**Step #1**

Now we will calculate all the values:

n = 20, since the coin was tossed 20 times

x = 5, as x is the number of success, in this question getting 5 heads is considered success

p = similarly the probability of success will be 0.5

q = 1-0.5 = 0.5

**Step#2**

We will now plug in the values into the formula above.

We can hence, conclude by saying, the probability of getting 5 heads by tossing a coin 20 times is 0.014.

Q. Out of 800 families, with 5 children each, how many families would you expect to have 3 boys.

*Solution:*

**Step #1**

We know that:

n = 5

x = 3

p = 0.5

q = 0.5

We will now plug the values into the formula to get:

We can hence conclude by saying,

The probability of having 3 boys is 0.3125

Now we have the probability, but we were asked how many families would you expect to have 3 boys, since the word expect is used in the question. We know that we have to calculate the mean

So we can say:

*E(X) = n p*

*= 800 x 0.3125*

*= 250*

Thus, there are 20 families who have 3 boys.

Q. About 75% of the population says that they never read books.

What is the probability that in a group of 4 randomly chosen people all read books.

*Solution:*

The probability of not read = 0.75 = q

The probability that read = 1- 0.75 = 0.25 = p

We have taken, because we have to find the probability that out of 4 all 4 of them read books.

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]]>The post Statistics appeared first on A Level Maths.

]]>Statistics is further divided into two branches:

- Descriptive statistics: It deals with describing a set of data graphically
- Inference statistics: This obtains information about a large set of data or future outcomes from a smaller sample

Statistics is closely linked to probability theory. We can use statistics to work out probabilities, Probability basically tells us about a chance, that a certain event will occur.

Moving on, as far as advanced level statistics is concerned, we will go through both these branches mentioned above. The topics we will cover in the statistics revision articles are:

**1. Averages**

Average can be described as the number that gives us a sense of a central tendency, or a number that is more representative of a set.

**2. Skewness**

Skewness measures the departure from symmetry.

- If the mean > median it indicates that the distribution is positively skewed.
- If the mean is < median it indicates that the distribution is negatively skewed.

**3. Box & Whisker diagrams**

A box and whiskers diagram is also known as box plot, it displays a summary of a set of data. Minimum, maximum, median, first quartile and third quartile, interquartile, upper limit and lower limit.

**4. Permutations & Combinations**

Both permutations and combinations are groups or arrangements of objects. With combinations, the order of the objects is insignificant, whereas in permutations the order of the objects makes a difference.

**5. Probability**

It is a measure of the likelihood that an event will happen.

**6. Bayes Theorem**

**7. Linear regression**

Regression line helps us connect two or more variables together, its equation is *y = ax + b*.

**8. Product moment correlation coefficient**

Formula for correlation:

**9. Measures of dispersion**

Measures the fluctuation/variation that is present in the data.

**10. Expectation & variance**

Expected value of a random variable:

Var(X), is defined by:

**11. Discrete random variables**

Discrete random variable are random variable that can take on distinct and separate variable.

**12. Normal distribution**

The normal distribution is a theoretical distribution of values.

.

**13. Central limit theorem**

The Central Limit Theorem (CLT) basically tells us that the sampling distribution of the sample mean is, at least approximately, normally distributed.

**14. Poisson distribution**

The probability function of the poisson distribution is:

.

**15. Uniform distribution**

In uniform distribution we should know that random variable is a continuous random variable. Probability density function *f*(*x*) to give .

**16. Binomial distribution**

The formula for binomial distribution is as follows:

**17. Geometric distribution**

The geometric distribution are the trails needed to get the first success in repeated and independent binomial trial.

**18. Histograms & Cumulative frequency**

A histogram show the distribution of numerical data. Cumulative frequency is accumulation of the frequencies

**19. Discrete uniform distribution**

The values of a discrete random variable are obtained by counting, thus making it known as countable. Uniform distribution simply means that when all of the random variable occur with equal probability.

**20. Continuous random variables**

If [a, b] are the domain of the continuous random variable function f and f(x) > 0 then:

**21. Normal approximations**

A normal approximation can be defined as a process where the shape of the binomial distribution is estimated by using the normal curve.

**22. One & two tailed tests**

A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both.

**23. Sampling**

Sampling is the selection of a subset of individuals from within a statistical population to estimate characteristics of the whole population.

**24. Estimators**

It is a rule, method, or criterion for arriving at an estimate of the value of a parameter.

**25. Confidence intervals**

A confidence interval is a range of values we are fairly sure our true value lies in.

**26. Hypothesis testing**

Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter.

**27. Random samples**

Random samples is known as the collection of independent and identically distributed random variables such as .

- https://en.wikipedia.org/wiki/Sampling_(statistics)
- https://www.investopedia.com/terms/o/one-tailed-test.asp
- https://studymoose.com/intro-to-statistics-essay
- http://www.mathscareers.org.uk/article/statistics-probability/
- http://lsc.cornell.edu/wp-content/uploads/2016/01/Why-study-statistics.pdf
- https://www.chegg.com/homework-help/definitions/normal-approximation-31

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