The probability is a very important branch of mathematics which in addition to statistics plays a vital role in maths, science, chemistry, psychology, geology, mining etc. Probability is a science of study of possibilities of happening an event. In probability, we usually calculate the chances of an event to be happening.

But what do we mean by an event ? First let us know what an experiment is. In probability theory, an experiment is a process which is supposed to have one or more results. For instance - Throwing a die, picking up a marble, flipping a coin are some simple experiments. An event is the set of all possible outcomes or results of such an experiment. There are different types of events - dependent events, independent events, mutually exclusive events etc. There is one more concept which we are going to discuss in this article in detail - **simple event**.

A simple event is an event that consists of exactly one outcome or we can say that, a simple event is the event of a single outcome. A probability value is assigned to each simple event using either relative frequency or personal probabilities. The notation P(E) is used to denote the probability of the event E to occur.

- Each probability of simple events must be lying between 0 and 1.
- The sum of the probabilities for all simple events in S equal to 1.

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A simple event is a unique possible outcome of a random circumstance. For an experiment, an event to be any collection of possible outcomes. Any particular outcome is known as a simple event. An event is a simple events or a subset of the sample space.

Probability is the chance or likelihood that an event will happen. It is the ratio of the number of favorable outcomes to the number of possible outcomes. The probability of an event E is equal to the sum of the probabilities of the sample events contained in E.

Probabilities for Equally Likely Simple Events

If there are m equally likely simple events in the sample space, then the probability for each is $\frac{1}{m}$. Supposed that a fair coin is tossed, then the sample space is {H, T}. And the probability is $\frac{1}{2}$ for each outcome.

Below you could see some examples of the probability:

### Solved Examples

**Question 1: **Toss two fair coins and record the outcome. Find the probability of finding the exactly one head in the two tosses.

** Solution: **

**Step 1:**

Sample space for the tossing two coins = {HH, HT, TH, TT}

Here the letters H and T means that head or a tail respectively. Since the sum of the four simple events must be 1, each must have probability P(E_{i}) = $\frac{1}{4}$.

**Step 2:**

The sample events in the sample space along with their equally likely probabilities.

Let P(A) = Probability of getting exactly one head.

=> P(A) = P(E_{2}) + P(E_{3})

= $\frac{1}{4} + \frac{1}{4}$

= $\frac{1}{2}$

**Question 2: **An urn contains 4 white and 3 white balls. Find the probability
distribution of the number of red balls in three draws, with
replacement, from the urn?

** Solution: **

Total number of balls = 4 + 3 = 7

Here p = $\frac{3}{7}$, q = $\frac{4}{7}$.

Let X denote a random variable which is the number of red balls obtained in three draws.

Clearly X takes the values 0, 1, 2 and 3

P(X = 0) = $\frac{4}{7}$ $\times$ $\frac{4}{7}$ $\times$ $\frac{4}{7}$ = $\frac{64}{343}$

P(X = 1) = 3 $\times$ ($\frac{3}{7}$ $\times$ $\frac{4}{7}$ $\times$ $\frac{4}{7}$)

= $\frac{144}{343}$

P(X = 2) = 3 $\times$ ($\frac{3}{7}$ $\times$ $\frac{3}{7}$ $\times$ $\frac{4}{7}$)

= $\frac{108}{343}$

P(X = 3) = $\frac{3}{7}$ $\times$ $\frac{3}{7}$ $\times$ $\frac{3}{7}$

= $\frac{27}{343}$

The probability distribution table is

Probability is the chance or likelihood that an event will happen. It is the ratio of the number of favorable outcomes to the number of possible outcomes. The probability of an event E is equal to the sum of the probabilities of the sample events contained in E.

Probabilities for Equally Likely Simple Events

If there are m equally likely simple events in the sample space, then the probability for each is $\frac{1}{m}$. Supposed that a fair coin is tossed, then the sample space is {H, T}. And the probability is $\frac{1}{2}$ for each outcome.

Below you could see some examples of the probability:

Sample space for the tossing two coins = {HH, HT, TH, TT}

Here the letters H and T means that head or a tail respectively. Since the sum of the four simple events must be 1, each must have probability P(E

The sample events in the sample space along with their equally likely probabilities.

Event |
Outcomes of the first coin |
Outcomes of the second coin |
P(E_{i}) |

E_{1} |
H |
H |
$\frac{1}{4}$ |

E_{2} |
H |
T |
$\frac{1}{4}$ |

E_{3} |
T |
H |
$\frac{1}{4}$ |

E_{4} |
T |
T |
$\frac{1}{4}$ |

Let P(A) = Probability of getting exactly one head.

=> P(A) = P(E

= $\frac{1}{4} + \frac{1}{4}$

= $\frac{1}{2}$

Total number of balls = 4 + 3 = 7

Here p = $\frac{3}{7}$, q = $\frac{4}{7}$.

Let X denote a random variable which is the number of red balls obtained in three draws.

Clearly X takes the values 0, 1, 2 and 3

P(X = 0) = $\frac{4}{7}$ $\times$ $\frac{4}{7}$ $\times$ $\frac{4}{7}$ = $\frac{64}{343}$

P(X = 1) = 3 $\times$ ($\frac{3}{7}$ $\times$ $\frac{4}{7}$ $\times$ $\frac{4}{7}$)

= $\frac{144}{343}$

P(X = 2) = 3 $\times$ ($\frac{3}{7}$ $\times$ $\frac{3}{7}$ $\times$ $\frac{4}{7}$)

= $\frac{108}{343}$

P(X = 3) = $\frac{3}{7}$ $\times$ $\frac{3}{7}$ $\times$ $\frac{3}{7}$

= $\frac{27}{343}$

The probability distribution table is

X | 0 | 1 | 2 |
3 |

P(X) | $\frac{64}{343}$ | $\frac{144}{343}$ | $\frac{108}{343}$ | $\frac{27}{343}$ |

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