Probability – Definition, Formula, Types, Solved Example Problems

# Probability – Definition, Formula, Types, Solved Example Problems

Table of Contents

Probability is a branch of mathematics that deals with the occurrence of random events. It is expressed from zero to one and predicts how likely events are to happen. In general, Probability is basically the extent to which something is likely to happen. You will learn about Probability Distribution where you will learn the possibility of outcomes for a random experiment.

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## Probability Definition

Probability is the measure of the likelihood of an event to occur. In the case of events, we can’t predict with total certainty. We can only predict the cancer of an event to occur i.e. how likely it is to happen. Probability ranges between 0 to 1 in which 0 indicates the event to be an impossible one and 1 indicates a certain event.

Example: For instance, when we toss a coin there are only two possibilities either head or tail(H,T). If we toss two coins in the air there are three possible outcomes that are both the coins show heads, both the coins show tails, one is head and the other is tail i.e. (H, H), (T, T), (H, T).

### Formula for Probability

Probability is defined as the possibility of an event to occur. The formula for Probability is given as the ratio of the number of favorable events to the total number of possible outcomes.

Probability of an event to happen = No. of Favourable Outcomes/ Total Number of Outcomes

This is the basic formula for Probability.

### Probability Tree

Tree Diagram helps to organize and visualize different possible outcomes. Branches and ends are the two main positions of the tree. Each branch Probability is written on the branch and the ends contain the final outcome. Tree Diagram helps you to figure out when to multiply and add.

### Types of Probability

There are three types of major probabilities. They are

• Theoretical Probability
• Experimental Probability
• Axiomatic Probability

Theoretical Probability: It depends on the possible chances of something to happen. Theoretical Probability mainly depends on the reasoning behind probability.

Experimental Probability: This kind of Probability depends on the observation of the experiment. Experimental Probability can be calculated on the number of possible outcomes to the total number of trials.

Axiomatic Probability: A Set of Rules or Axioms are Set applies to all types. Using the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified.

Conditional Probability is nothing but the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

### Probability of an Event

Let us consider an Event E that occurs in r ways out of n possible ways. The probability of happening an event or its success is given by

P(E) = r/n

The probability of an event or its failure is given by

P(E’) = (n-r)/n = 1-(r/n)

E’ represents the event will not occur.

Therefore, we can say that

P(E)+P(E’) = 1

### What are Equally Likely Events?

If the events have the same theoretical probability of happening then they are called Equally Likely Events. Results of Sample Space are said to be equally likely if all of them have the same probability of occurring. Below are some examples of Equally Likely Events.

• Getting 2 or 3 on throwing a die.
• Getting 1, 3, 4 on throwing a die

are all Equally Likely Events since the Probabilities of Each Event are Equal.

### Complementary Events

In the Case of Such Events, there will only be two outcomes that state whether an event will occur or not. The complement of an event occurring is the exact opposite that the probability of an event is not occurring.

• It may or may not rain today
• Winning a lottery or not.
• You win the lottery or you don’t.

### Probability Density Function

It is a probability function that represents the density of continuous random variables lying between a certain range of values. Standard Normal Distribution is used to create a database or statistics that are used in science to represent the real-valued variables whose values are unknown.

### Additive Law of Probability

If E1 and E2 be any two events (not necessarily mutually exclusive events), then P(E1 ∪ E2) = P(E1) + P(E2) – P(E1 ∩ E2)

### Probability Terms

Some of the Important Probability Terms are discussed here

Sample Space: Set of all possible outcomes that occur in any trail.

Example:

Tossing a Coin, Sample Space (S) = {H, T}

When you Roll a Die Sample Space (S) = {1, 2, 3, 4, 5, 6}

Sample Point: It is one of the Possible Outcome.

Example:

In a deck of cards, 3 of hearts is a sample point.

Experiment or Trial: Series of Actions where outcomes are always uncertain.

Example: Tossing a Coin, Choosing a Card from a Deck of Cards, Throwing a Dice.

Event: Single outcome of an experiment.

Example: Getting Tails while Tossing a Coin is an Event.

Outcome: Possible Result of an Experiment.

Head is a possible outcome when a coin is tossed.

Impossible Event: The Event can’t happen

While Tossing a Coin it is impossible to get head and tail at the same time.

### Solved Examples on Probability

1. Find the Probability of getting 2 on rolling a die?

Solution:

Sample Space = {1, 2, 3, 4, 5, 6}

Number of Favourable Events = 1 i.e. {2}

Total Number of Outcomes = 6

Probability P = 1/6

Therefore, Probability of getting 2 on rolling a die is 1/6.

2. Two dice are rolled find the Probability that the Sum is

equal to 1

equal to 5

equal to 8

Solution:

In order to find the probability whose sum is equal to 5 we need to figure out the Sample Space

S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }
(i) Let E be the event whose sum is equal to 1. There are no possibilities to get the sum 1 when a dice is rolled.
P(E) = n(E)/n(S)
= 0/36
= 0
(ii) Let E be the event whose sum is equal to 5. The possibilities of getting 5 when a dice is rolled is (1,4) (4, 1) (2, 3) (3, 2)
Hence, P(E) = n(E)/n(S)
= 4/36
= 1/9
(iii) Let E be the event whose sum is equal to 8. The Possibilities of getting 8 when a dice is rolled is (2, 6) (3, 5) (4, 4) (5, 3), (6, 2)
Hence, P(E) = n(E)/n(S)
=5/36
3. A dice is thrown 35 times and 4 appeared 16 times. Now, in a random throw of a dice, what is the probability of getting a 4?
Solution:
Total number of trials = 35
Number of times 4 appeared = 16
Probability of getting 4 = Number of times 4 appeared/Total Number of Trails
= 16/35
The probability of getting 4 when a dice is thrown is 16/35.
4. Draw a random card from a pack of cards. What is the probability that the card drawn is an ace card?
Solution:
Total Number of Outcomes = 52
No. of Aces in a deck of cards = 4
Probability of drawing an ace = 4/52
= 1/13
Therefore, the probability of drawing an ace from a deck of cards is 1/13.

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