In the previous section, we completed a discussion on statistics. In this chapter, we will see probability.
In our earlier classes, we have seen the basics of probability. The links to those notes are given below:
♦ Probability part I consists of chapters 1.5, 1.6, . . . up to 1.9
♦ probability part II consists of chapters 28, 28.1, . . . up to 28.4
♦ probability part III consists of chapters 36, 37.1 and 36.2
The reader must have a thorough knowledge on the above three parts. In our present discussion, we will see some advanced techniques.
First we must become familiar with four basic terms:
(i) Random experiments, (ii) Outcomes (iii) Sample space (iv) Sample point.
Random experiments
This can be explained in 4 steps:
1. Consider an experiment in which a student is analyzing the sum of interior angles of various triangles.
• He takes the first triangle and measure it’s three interior angles. He finds that the sum of those three angles is 180o
• He takes the second triangle and measure it’s three interior angles. He finds that the sum of those three angles is 180o
• He can take any number of triangles. The sum will be always 180o.
• So the experiment of “analyzing sum of interior angles of triangles” will always give a fixed result. We can predict the result. We can predict that, the sum will be 180o
2. Consider the experiment of tossing a coin.
• A student tosses a coin. The result can be head or tail. We cannot predict the result.
• The student tosses the coin again. This time also, the result can be head or tail. We cannot predict the result.
• In this experiment,
♦ There are two possible results: head and tail.
♦ It is not possible to predict the result.
• Such experiments are called random experiments.
• Note that, if there is only one possible result, a need for prediction will not arise.
• We can write:
An experiment is a random experiment if it satisfies two conditions:
(i) It has more than one possible result.
(ii) It is not possible to predict the result in advance.
3. Based on the above steps, we can write:
The experiment of rolling a die is a random experiment.
• This is because, there are six possible results. Also, it is not possible to predict the result.
4. In this chapter, whenever we mention the word “experiment”, it will be a random experiment.
Outcomes
This can be explained in 4 steps:
1. Consider the experiment of rolling a die.
2. The possible results are: 1, 2, 3, 4, 5 and 6
3. Each of the above possible results is called an outcome of the experiment.
4. So we can write:
♦ Outcomes of an experiment
♦ are
♦ possible results of that experiment.
Sample space
This can be explained in 4 steps:
1. Consider the experiment of rolling a die.
2. The outcomes are: 1, 2, 3, 4, 5 and 6
3. Using these outcomes, we can form a set S:
S = {1, 2, 3, 4, 5, 6}
• This set is called sample space of the experiment.
4. So we can define sample space in 3 steps:
(i) Sample space of an experiment, is a set
(ii) All outcomes of that experiment will be elements of that set.
(iii) This set is denoted by the letter ‘S’.
Sample point
This can be explained in 2 steps:
1. Each element of S is called a sample point.
2. But we know that, each element is an outcome.
• So we can write:
Each outcome is a sample point.
Let us see some solved examples:
Solved Example 16.1
Two different coins are tossed once. Find the sample space.
Solution:
1. Given that the coins are different.
♦ So the head of one coin will be different from the head of the other coin.
♦ Also, the tail of one coin will be different from the tail of the other coin.
• Because of this difference, we will give specific names:
♦ Head of first coin can be named as H1.
♦ Head of the other coin can be named as H2.
♦ Tail of first coin can be named as T1.
♦ Tail of the other coin can be named as H2.
2. Let us write the possible outcomes:
(i) Outcome 1:
♦ First coin gives H1.
♦ The other coin gives H2.
• Note that, in this outcome, T1 and T2 are not possible. For example, the first coin landed with H1 on the upper side. So T1 will not be visible.
• We can write this outcome as: (H1,H2)
(ii) Outcome 2:
♦ First coin gives H1.
♦ The other coin gives T2.
• Note that, in this outcome, T1 and H2 are not possible.
• We can write this outcome as: (H1,T2)
(iii) Outcome 3:
♦ First coin gives T1.
♦ The other coin gives H2.
• Note that, in this outcome, H1 and T2 are not possible.
• We can write this outcome as: (T1,H2)
(iv) Outcome 4:
♦ First coin gives T1.
♦ The other coin gives T2.
• Note that, in this outcome, H1 and H2 are not possible.
• We can write this outcome as: (T1,T2)
3. Based on the above outcomes, we can write the sample space as:
S = {(H1,H2), (H1,T2), (T1,H2), (T1,T2)}
Solved Example 16.2
Two different dice (one blue and the other red) are rolled once. Find the sample space.
Solution:
1. We have already seen this type of problem in our earlier classes (see solved example 1.14 in section 1.9).
2. In our present case, we are asked to write the sample space.
• For presenting the sample space in a “convenient and easy to understand” manner, we use ordered pairs (x,y)
• If we take the blue die as the first die, then:
♦ All values associated with blue die are x.
♦ All values associated with red die are y.
3. Based on such a notation, the sample space for our present case can be written as:
S ={
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
• There are 36 elements in S.
Solved example 16.3
In each of the following experiments, write the sample space.
(i) A boy has a 1 rupee coin, a 2 rupee coin and a 5 rupee coin in his pocket. He takes two coins out of his pocket, one after the other.
(ii) A person is noting down the number of accidents along a busy highway during a year.
Solution:
Part (i):
1. We will use the form of ordered pairs (x,y)
♦ x is related to the first coin taken out.
♦ y is related to the second coin taken out.
2. Let us write the outcomes:
(i) First coin is 1, second coin is 2.
We write this as (1,2).
(ii) First coin is 1, second coin is 5.
We write this as (1,5)
(iii) First coin is 2, second coin is 1.
We write this as (2,1)
(iv) First coin is 2, second coin is 5.
We write this as (2,5)
(v) First coin is 5, second coin is 1.
We write this as (5,1)
(vi) First coin is 5, second coin is 2.
We write this as (5,2)
3. So the sample space can be written as:
S = {(1,2), (1,5), (2,1), (2,5), (5,1), (5,2)}
• There are 6 elements in S.
Part (ii):
1. The number of accidents in a year may be 0 (if there are no accidents), or any positive integer.
2. So the sample space can be written as:
S = {0, 1, 2, 3, . . . }
Solved example 16.4
A coin is tossed. If it shows head, we draw a ball from a bag containing 3 blue and 4 white balls. If it shows tail, we through a die. Write the sample space of this experiment.
Solution:
1. We will use the form of ordered pairs (x,y)
♦ x is related to tossing of coin.
♦ y is related to drawing a ball or rolling the die.
2. Let us write the out comes:
(i) The coin shows heads. The ball drawn is the first blue ball.
• We write this as (H,B1).
(ii) The coin shows heads. The ball drawn is the second blue ball.
• We write this as (H,B2).
(iii) The coin shows heads. The ball drawn is the third blue ball.
• We write this as (H,B3).
(iv) The coin shows heads. The ball drawn is the first white ball.
• We write this as (H,W1).
(v) The coin shows heads. The ball drawn is the second white ball.
• We write this as (H,W2).
(vi) The coin shows heads. The ball drawn is the third white ball.
• We write this as (H,W3).
(vii) The coin shows heads. The ball drawn is the fourth white ball.
• We write this as (H,W4).
(viii) The coin shows tails. The die shows 1.
We write this as (T,1).
(ix) The coin shows tails. The die shows 2.
We write this as (T,2).
(x) The coin shows tails. The die shows 3.
We write this as (T,3).
(xi) The coin shows tails. The die shows 4.
We write this as (T,4).
(xii) The coin shows tails. The die shows 5.
We write this as (T,5).
(xiii) The coin shows tails. The die shows 6.
We write this as (T,6).
3. So the sample space can be written as:
S = {(H,B1), (H,B2), (H,B3), (H,W1), (H,W2), (H,W3), (H,W4), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
• There are 13 elements in S.
Solved Example 16.5
Consider the experiment in which a coin is tossed repeatedly until a head comes up. Write the sample space.
Solution:
1. Let us write the outcomes:
(i) Head is obtained in the first toss.
We write this as H.
(ii) Head is obtained only in the second toss.
We write this as TH.
(iii) Head is obtained only in the third toss.
We write this as TTH.
(iv) Head is obtained only in the fourth toss.
We write this as TTTH.
so on . . .
2. So the sample space can be written as:
S = {H, TH, TTH, TTTH, . . . }
Link to a few more solved examples is given below:
In the next section, we will see Event.
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