Probability | Material

Probability:

Probability is a measure of uncertainity (an event may happen or may not happen).

Probability is classified into the following two types.

(i) Empirical probability

(ii) theoretical probability.

Empirical Probability Or Experimental Probability

Empirical Probability is the probability of events on the basis of the results obtained through actual experiments.

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Theoretical probability Or Classical Probability:

Theoretical probability Or Classical Probability is the probability of events based on the results obtained from theoretical approach.

Theoretical Approach

Here, we try to predict the outcomes without performing an actual experiment.

We assume that the outcomes of an experiment are equally likely.

We find that the experimental probability of an event approaches its theoretical probability if the number of trials of an experiment is very large.

Key Terms

Experiment:

Experiment is An activity that causes some well defined outcomes.

Random Experiment

Random experiment is an experiment that

may not give the same result on repitition

under identical conditions,

Trial: Each repition of the activity is called A Trial.

Outcome

Each possible result of a random experiment or a Trial is called its outcome.

Sample Space

The collection of all possible outcomes in a random experiment is called Sample space.

Or

The total number of possible outcomes of a random experiment.

As the number of trials in an experiment increases we may expect the empirical and theoretical probabilities to be nearly the same.

Event:

Any subset of smaple space is called an Event.

Types of Events

Compound Event An event connected to a random experiment is a compound event if it is obtained by combining two or more elementary events connected to the random experiment.

Occurrence of an event An event corresponding to a random experiment is said to occur if any one of the elementary events corresponding to the event is the outcome.

Impossible Events The event which never occurs is an impossible event.

So the probability ofan impossible eventis always zero.

Sure Event The event which certainly occurs is a sure event.In general, it is truethat for anevent E, ( ̅) ( ) Here theevent E is representing “not E”. This is called the compound ofthe event ‘E’.So ‘E’ and E are complementary events.

Theoretical Probability of An Event:

If there are n events associated with a random experiment and m of them are favorable to an event E, then the probability of the event E is denoted by P(E) and is calculated using the follwoing formula.

i.e.,

• Probability of an event E lies between 0 and 1.
• If P (E) =1, then event E is called a certain event or sure event.
• If P (E) =0, then E is called an impossible event.
• Number of non-occurrence of event E = n – m

Where the event

representing non-occurrence of event E, is called the complement of the event E.

Hence, E and

are complementary events.

Sum of the probabilities of all the elementary events of an experiment is

i.e.,

P (E1) + P (E2) + P (E3) … + P (En) = 1

Playing Cards:

A pack of cards consists of four suits. They are

Each suit consists of 13 cards, nine cards numbered 2, 3, 4 ….10, an Ace (B) a Jack (J), a Queen (Q), and a King (K).

Spades and Clubs are black in colour.

Hearts and Diamonds are red in colour.

So there are 26 black cards and 26 red cards.

King, queen and jack are called face cards.

There are totally 12 (4 x3) face cardsin a packof 52 cards.

I.e. in eachsuit we have3 face cards.

Coins:

A coin has two sides namely head and tail.

In the experiment of tossing a coin for once, there are 2 possible outcomes. They are 1 head, 1 tail.

P (Head) = 1/2 = P (Tail)

Dice:

A die is a well-balanced cubewith six facesnumbered from 1 to 6.

Dice is the pluralform. There are six equally likely outcomes 1, 2,3, 4, 5,6 in a single throw. Geometric Probability:

If the total number of outcomes of a trial in a random experiment is infinite, then the above definition is not sufficient to find the probability of an event.

In such cases, the definition of probability is modified and probability so obtained is called Geometric Probability.

The geometric probability p of an event is given by