Probability Formula and Variance Formula

Probability Formula

We all are knowing about occurrences of various events and their chances to occur. Probability is a wonderfully usable and applicable field of mathematics. The theory of probability began in the 17th century in France by two mathematicians Blaise Pascal and Pierre de Fermat. In this article, we will mainly be focusing on probability formula and examples. Let us get started!

Probability Formula

What is Probability?

We are usually about a weather forecast at the end of a news bulletin on TV or we read about the weather conditions of our city for the next few days in the newspaper. They specifically use the term which is the probability.

Probability is the term used in math as well as in statistics very frequently. It is defined as the measure of chances for the occurrence of some event. It also provides the estimation of uncertainty of any event. For example, getting head or tail during toss of a coin is having a chance of 12 for both.

There are some terms used in the study of probability are as follows:

1. Experiment: It is any phenomenon like rolling a dice, tossing a coin, drawing a card from a well-shuffled deck, etc.
2. Outcome: It is the result of any event such as number appearing on a dice, side of a coin etc.
3. Sample Space: It is the set of all possible outcomes.
4. Event: It is the combination of possible outcomes or the subset of sample space. For example getting an even number on rolled dice, getting a head or tail on a flipped coin, taking out a king of any card suit.
5. Probability Function: It is a function for giving the probability for each outcome.
6. Odds in favor of the event: It is the ratio of the number of ways that an outcome can occur to the number of ways it cannot occur.
7. Odds against the event: It is the ratio of the number of ways that an outcome cannot occur to the number of ways it can occur.

Probability Formulae

Probability = NumberofaFavourableoutcomeTotalnumberofoutcomes

i.e.      P= N(E)N(S) 

Here,

P	probability
E	event
S	sample space.
n( E)	the count of favourable outcomes
n(S)	the size of the sample space.

P is the probability, E is some event and S is its sample space.

Where, n( E) = the count of favorable outcomes

and n(S) = the size of the sample space.

Solved Examples

To understand the above formula let us have some examples.

Q: Probability for getting an even number on the front face of a rolling dice.

Solution: Here sample space (S) = {1, 2, 3, 4, 5, 6} i.e. all possible outcomes.

And, event (E) = {2, 4, 6} i.e. even number occurrence.

Thus n (S) = 6 and n (E) = 3

Using this in the probability formula, we get:

P = 36 = 12 = 0.5

Therefore the chances of getting an even number upon rolling a dice is 0.5

Q: Find the probability of getting HEAD at least once on tossing a coin twice.

Solution: Here sample space (S) = {HH, HT, TH, TT}

H denotes Head and T denotes Tail.

So, favourable event (E) = {HH, HT, TH}

Thus n (S) = 4 and n (E) = 3

Using these values in probability formula, we get:

P = 34 = 0.75

Hence, the chances of getting at least one HEAD on tossing a coin twice are 0.75

Variance Formula

Variance is a measure of how data points differ from the mean value. According to the simple terms, it is a measure of how far a set of data i.e. numbers are spread out from their mean i.e. average value. Variance is a statistical measure that tells us how measured data vary from the average value of the set of data. In other words, a variance is the mean of the squares of the deviations from the arithmetic mean of a data set. In this article, we will discuss the variance formula.

Variance Formula

What is a Variance?

Variance is used in how far a set of numbers are spread out. This is different from finding the average, or the mean, of numbers. For example, take a look at the following numbers: 12, 8, 10, 10, 8, 12. If we add these numbers together and divide by the total numbers in the data set, which in this case is 6, you will get an average of 10. Notice that these numbers are all pretty close to the number 10.

Now take a look at this set of data: 28, 4, 6, 4, 2, 16. We can notice that there is a greater difference between the numbers in the second set of data versus the first set of data. Although both sets of data have an average of ten. We show these differences in data by using variance.

There are two types of variance, one is population and the other is a sample. Population is having all members of a specified group. If we were to collect data on just the members of our household, then everyone living in would be considered the population. A sample is a part of the population used to describe the whole group.

If we collect data on the members of our household and only collect data about two members out of the five members. Then this will be considered a sample. Other examples of population and samples would be the total members of a school versus only the members of a class in the school or a random selection of 50 members of a school, which will also be a sample.

Variance Formula

For the purpose of solving questions, it is,

Var(X)=E[(X−μ)2]

Var(X) will represent the variance.

This means that variance is the expectation of the deviation of a given random set of data from its mean value and then squared.

Here, X is the data,

µ is the mean value equal to E(X), so the above equation may also be expressed as,

Var(X)=E[X−E(X)2]

OrVar(X)=E(X2)−E(X)2

Solved Examples

Example: Find the variance of the numbers 3, 8, 6, 10, 12, 9, 11, 10, 12, 7.

Solution:

Step 1: First compute the mean of the 10 values given.

X¯=3+8+6+10+12+9+11+10+12+710

= 8810

= 8.8

Step 2: Make a table as following with three columns, one for the X values, the second for the deviations and the third for squared deviations.

Value (X)	X–X¯	(X−X¯)2
3	-5.8	33.64
8	-0.8	0.64
6	-2.8	7.84
10	1.2	1.44
12	3.2	10.24
9	0.2	0.04
11	2.2	4.84
10	1.2	1.44
12	3.2	10.24
7	-1.8	3.24
Total	0	73.6

Step 3:

As the data is not given as sample data, thus we use the formula for population variance.

= 73.610

= 7.36