Class-IX Unit-14 Probability

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What is Probability ?

Probability is the measure of the likeliness that an event will occur.Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty)

Why we need Probability and what is the use of it?

It is widely used in the study of Mathematics, Statistics, Gambling, Physical sciences, Biological sciences, Weather forecasting, Finance etc. to draw conclusions. Insurance companies uses this to decide on financial policies

There are few terms related to Probability which are defined below

Randomness

In mathematics, When next outcome of the experiment can not be determined then we say it is a random experiment e.g. Consider the dice. When we throw the dice, we cannot determine what number will come Since we cannot predict the next outcome, we may say it is a random experiment

Trial

A trial is an action which results in one or several outcomes, for example each toss of the coin and each throw of the die are called trials

Independent Trial

:Successive trials of some random event for example tosses of a coin,throws of a die are said to be independent if the outcome of any one trial does not impact the outcomes of any others.

Experiment

Experiment and trial are same thing as such.An experiment is a situation involving chance or probability that leads to results called outcomes.But sometimes we use experiment to refer to whole large number of trials.

Event

An event is a possible outcome of the Experiment. Like head coming in a toss

Sample space

It is a set of all possible outcomes of an experiment.

e.g. when we coin is tossed,the possible outcome are Head and Tail.So sample space is Head and tail 

Empirical Probability:

1) Experimental or empirical probability is an estimate that an event will happen based on how often the event occurs after performing an experiment in a large number of trials. 
2) It is a probability of event which is calculated based on experiments

3) Empirical probability depends on experiment and different will get different values based on the experiment

Some Important points

1) If the event A, B, C covers the entire possible outcome in the experiment. Then,

P (A) +P (B) +P(C) =1
2) The probability of an event (U) which is impossible to occur is 0. Such an event is called an impossible event
P (U)=0
3) The probability of an event (X) which is sure (or certain) to occur is 1. Such an event is called a sure event or a certain event
P(X) =1
4) Probability of any event can be as
0 ≤ P(E) ≤ 1 

Solved Examples

A coin is tossed 1000 times; we get 499 times head and 501 times tail

1) What is the empirical probability of getting head ?
2) What is the empirical probability of getting tail?
3) Does the sum of above two probability equals 1?
Solution: empirical probability of getting head=(No of trails which heads came)/(Total Number of trials)
empirical probability of getting head=499/1000

So empirical or experimental probability of getting head P(H) is calculated as
=.499
empirical probability of getting tail =(No of trails which tails came)/(Total Number of trials)
empirical probability of getting head=501/1000
So empirical or experimental probability of getting head P(T) is calculated as

=.501

Now P(H) +P(T)= .499+.501=1
As tail and head are the only possible outcome,the sum of probabilities is 1