# Math՚s: Theory of Probability: Random Experiment, Probability of an Event (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Theory of Probability

• The theory of probability is a branch of mathematics which deals with the analysis of random phenomena or situations involving uncertainty.
• The probability theory is applied in many fields such as insurance industry, meteorological department, gambling, biology, financial markets etc. Insurance Industry use probability theory to determine the prices of policies. Similarly, meteorologists use probability theory to predict weather conditions in future periods.

## Random Experiment and Its Outcome

An experiment which has more than one possible outcome all of which are already known, and any particular outcome is uncertain is known as a random experiment.

• For Example- If a die is rolled, it may show any of the numbers from to . Thus, rolling a die is a random experiment as it has six possible outcomes and each outcome is uncertain.
• Similarly, tossing a coin, drawing a card from a deck of cards are also random experiments.

## Probability of an Event

• One or more outcomes constitute an event of an experiment. For example, In case of tossing a coin, “the coin shows up a head” or “the coin shows up a tail” each is an event, the first one corresponds to the outcome and the other to the outcome .
• The probability of an event is the measure of the likelihood of the occurrence of an event in an experiment. The probability of an event is defined as the number of outcomes favourable to divided by the total number of outcomes in an experiment.

i.e..

The probability of any event ranges from to . i.e.. .

Example- A die is thrown once. What is the probability of getting an even number?

Solution- Let be the event “getting an even number” .

Possible outcomes of the experiment are:

Outcomes favourable to E:

The probability of getting an even number

## Types of Events

• Certain Event- An event which is sure to occur is a certain event. The probability of a certain event is equal to . For Ex- In throwing of a die, the event of “getting a number less than ” is a certain event since every number on a die is always less than .
• Impossible Event- When there is no chance of the event occurring, it is said to be an impossible event. The probability of an impossible event is equal to . The event of “getting a number greater than ” on a die is an impossible event.
• Equally Likely Events- When two or more events have the same likelihood of occurrence, they are known as equally likely. For Ex- in the tossing of a coin, the two events “heads” and “tails” are equally likely as both have 50 % chance of occurrence.
• Complementary Events- For an event , the non-occurrence of the event is called its complementary event. Basically, complementary events are the events which cannot occur at the same time. The sum of probabilities of complementary events sum up to . When a die is thrown, “getting an odd face” and “getting an even face” are complementary events.

Example- If , what is the probability of the complement of ?

Solution- The complement of will be . We know that the sum of complementary events equals

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