In everyday life, we come across statements such as

(1) It will `text (probably )` rain today.

(2) I `text (doubt)` that he will pass the test.

(3) `text (Most probably ) ` , Kavita will stand first in the annual examination.

(4) `text ( Chances )` are high that the prices of diesel will go up.

(5) There is a 50-50 ` text (chance)` of India winning a toss in today’s match.

The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the statements above involve an element of uncertainty. For example, in (1), ‘probably rain’ will mean it may rain or may not rain today.

We are predicting rain today based on our past experience when it rained under similar conditions. Similar predictions are also made in other cases listed in (2) to (5).

The uncertainty of ‘probably’ etc can be measured numerically by means of ‘probability’ in many cases.

Though probability started with gambling, it has been used extensively in the fields of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc.

In earlier classes, you have had a glimpse of probability when you performed experiments like tossing of coins, throwing of dice, etc., and observed their outcomes. You will now learn to measure the chance of occurrence of a particular outcome in an experiment.

`text ( Activity 1 : )` (i) Take any coin, toss it ten times and note down the number of times a
head and a tail come up. Record your observations in the form of the following table



Write down the values of the following fractions:




and



(ii) Toss the coin twenty times and in the same way record your observations as above. Again find the values of the fractions given above for this collection of observations.

(iii) Repeat the same experiment by increasing the number of tosses and record the number of heads and tails. Then find the values of the corresponding fractions.

You will find that as the number of tosses gets larger, the values of the fractions come closer to 0.5. To record what happens in more and more tosses, the following group activity can also be performed:

`text (Acitivity 2 : )` Divide the class into groups of 2 or 3 students. Let a student in each group toss a coin 15 times.

Another student in each group should record the observations regarding heads and tails. [Note that coins of the same denomination should be used in all the groups. It will be treated as if only one coin has been tossed by all the groups.]

Now, on the blackboard, make a table like Table 15.2. First, Group 1 can write down its observations and calculate the resulting fractions.

Then Group 2 can write down its observations, but will calculate the fractions for the combined data of Groups 1 and 2, and so on. (We may call these fractions as cumulative fractions.) We have noted the first three rows based on the observations given by one class of students.



What do you observe in the table? You will find that as the total number of tosses of the coin increases, the values of the fractions in Columns (4) and (5) come nearer and nearer to `0.5`.

`text (Activity 3 : )` (i) Throw a die* 20 times and note down the number of times the numbers

*A die is a well balanced cube with its six faces marked with numbers from 1 to 6, one number on one face. Sometimes dots appear in place of numbers.


1, 2, 3, 4, 5, 6 come up. Record your observations in the form of a table, as in Table 15.3:



Find the values of the following fractions:



(ii) Now throw the die 40 times, record the observations and calculate the fractions
as done in (i).

As the number of throws of the die increases, you will find that the value of each fraction calculated in (i) and (ii) comes closer and closer to `1/6`.


To see this, you could perform a group activity, as done in Activity 2. Divide the students in your class, into small groups.
One student in each group should throw a die ten times. Observations should be noted and cumulative fractions should be calculated.


The values of the fractions for the number 1 can be recorded in Table 15.4. This table can be extended to write down fractions for the other numbers also or other tables of the same kind can be created for the other numbers.



The dice used in all the groups should be almost the same in size and appearence. Then all the throws will be treated as throws of the same die.

What do you observe in these tables?
You will find that as the total number of throws gets larger, the fractions in Column (3) move closer and closer to `1/6` .

`text ( Activity 4 : )` (i) Toss two coins simultaneously ten times and record your observations in the form of a table as given below:



Write down the fractions:



Calculate the values of these fractions.

Now increase the number of tosses (as in Activitiy 2). You will find that the more the number of tosses, the closer are the values of `A, B` and `C` to `0.25, 0.5` and `0.25`, respectively.

In Activity `1`, each toss of a coin is called a trial. Similarly in Activity `3`, each throw of a die is a trial, and each simultaneous toss of two coins in Activity `4` is also a trial.

So, a trial is an action which results in one or several outcomes. The possible outcomes in Activity 1 were Head and Tail; whereas in Activity `3`, the possible outcomes were `1, 2, 3, 4, 5` and `6`.

In Activity 1, the getting of a head in a particular throw is an event with outcome `text ( ‘head’ )`. Similarly, getting a tail is an event with outcome `text ( ‘tail’ )` . In Activity `2`, the getting of a particular number, say `1`, is an event with outcome `1`.

If our experiment was to throw the die for getting an even number, then the event would consist of three outcomes, namely, `2, 4` and `6`.

So, an event for an experiment is the collection of some outcomes of the experiment. In Class X, you will study a more formal definition of an event.

So, can you now tell what the events are in Activity 4?

With this background, let us now see what probability is. Based on what we directly observe as the outcomes of our trials, we find the experimental or empirical probability.

Let `n` be the total number of trials. The empirical probability `P(E)` of an event `E` happening, is given by

`P(E) = ( text (Number of trials in which the event happened ) )/( text (The total number of trials) )`

In this chapter, we shall be finding the empirical probability, though we will write ‘probability’ for convenience.

Let us consider some examples.

To start with let us go back to Activity `2`, and Table 15.2. In Column (4) of this table, what is the fraction that you calculated? Nothing, but it is the empirical probability of getting a head.

Note that this probability kept changing depending on the number of trials and the number of heads obtained in these trials. Similarly, the empirical probability
of getting a tail is obtained in Column (5) of Table 15.2. This is `12/15` to start with, then it is `2/3` , then `28/45` , and so on .

So, the empirical probability depends on the number of trials undertaken, and the number of times the outcomes you are looking for coming up in these trials.

`text ( Activity 5 : )` Before going further, look at the tables you drew up while doing Activity 3. Find the probabilities of getting a 3 when throwing a die a certain number of times. Also, show how it changes as the number of trials increases.