In our day to day life, we perform many activities which have a fixed result no matter any number of times they are repeated. For example given any triangle, without knowing the three angles, we can definitely say that the sum of measure of angles is \(180^{\circ}\).
We also perform many experimental activities, where the result may not be same, when they are repeated under identical conditions. For example, when a coin is tossed it may turn up a head or a tail, but we are not sure which one of these results will actually be obtained. Such experiments are called random experiments.
An experiment is called random experiment if it satisfies the following two conditions:
Check whether the experiment of tossing a die is random or not?
In this chapter, we shall refer the random experiment by experiment only unless stated otherwise.
Definition of Random Experiment
An experiment whose outcome cannot be predicted with certainty, is called a random experiment.
\( \quad \quad \quad \quad \quad \quad \text { Or }\)
If in each trial of an experiment, which when repeated under identical conditions, the outcome is not unique but the outcome in a trial is one of the several possible outcomes, then such an experiment is known as a random experiment.
For example,
Outcomes and sample space
Definition of Outcome: A possible result of a random experiment is called its outcome.
Sample Space Definition: The set of all possible outcomes of a random experiment is called the sample space associated with the experiment. Sample space is denoted by the symbol S. Each element of the sample space is called a sample point.
Consider the experiment of rolling a die. The outcomes of this experiment are 1, \(2,3,4,5\), or 6 , if we are interested in the number of dots on the upper face of the die.
The set of outcomes \(\{1,2,3,4,5,6\}\) is called the sample space of the experiment.
Each element of the sample space is called a sample point. In other words, each outcome of the random experiment is also called sample point.
Let us now consider some examples.
Example 1: Two coins (a one rupee coin and a two rupee coin) are tossed once. Find a sample space.
Solution: Clearly the coins are distinguishable in the sense that we can speak of the first coin and the second coin. Since either coin can turn up Head \((H)\) or Tail \((T)\), the possible outcomes may be
Heads on both coins \(=( H , H )= HH\)
Head on first coin and Tail on the other \(=( H , T )= HT\)
Tail on first coin and Head on the other \(=( T , H )= TH\)
Tail on both coins \(=( T , T )= TT\)
Thus, the sample space is \(S =\{ HH , HT , TH , TT \}\)
Note: The outcomes of this experiment are ordered pairs of \((H)\) and Tail \((T)\). For the sake of simplicity the commas are omitted from the ordered pairs.
Example 2: Find the sample space associated with the experiment of rolling a pair of dice (one is blue and the other red) once. Also, find the number of elements of this sample space.
Solution: Suppose 1 appears on blue die and 2 on the red die. We denote this outcome by an ordered pair \((1,2)\). Similarly, if ‘ 3 ‘ appears on blue die and ‘ 5 ‘ on red, the outcome is denoted by the ordered pair \((3,5)\).
In general each outcome can be denoted by the ordered pair \((x, y)\), where \(x\) is the number appeared on the blue die and \(y\) is the number appeared on the red die. Therefore, this sample space is given by
\(S =\{(x, y): x\) is the number on the blue die and \(y\) is the number on the red die \(\}\). The number of elements of this sample space is \(6 \times 6=36\) and the sample space is given below:
\(
\begin{aligned}
& \{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6) \\
& (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6) \\
& (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}
\end{aligned}
\)
Example 3: In each of the following experiments specify appropriate sample space
(i) A boy has a 1 rupee coin, a 2 rupee coin and a 5 rupee coin in his pocket. He takes out 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: (i) Let Q denote a 1 rupee coin, H denotes a 2 rupee coin and R denotes a 5 rupee coin. The first coin he takes out of his pocket may be any one of the three coins \(Q , H\) or R . Corresponding to Q , the second draw may be H or R . So the result of two draws may be QH or QR . Similarly, corresponding to H , the second draw may be Q or R .
Therefore, the outcomes may be HQ or HR. Lastly, corresponding to R, the second draw may be H or Q .
So, the outcomes may be RH or RQ.
Thus, the sample space is \(S =\{ QH , QR , HQ , HR , RH , RQ \}\)
(ii) The number of accidents along a busy highway during the year of observation can be either 0 (for no accident ) or 1 or 2 , or some other positive integer. Thus, a sample space associated with this experiment is \(S=\{0,1,2, \ldots\}\)
Example 4: A coin is tossed. If it shows head, we draw a ball from a bag consisting of 3 blue and 4 white balls; if it shows tail we throw a die. Describe the sample space of this experiment.
Solution: Let us denote blue balls by \(B _1, B_2, B_3\) and the white balls by \(W _1, W_2, W_3, W_4\). Then a sample space of the experiment is
\(
S =\left\{ HB _1, HB _2, HB _3, HW _1, HW _2, HW _3, HW _4, T 1, T 2, T 3, T 4, T 5, T 6\right\} .
\)
Here \(HB _i\) means head on the coin and ball \(B _i\) is drawn, \(HW _i\) means head on the coin and ball \(W _i\) is drawn. Similarly, T \(i\) means tail on the coin and the number \(i\) on the die.
Example 5: Consider the experiment in which a coin is tossed repeatedly until a head comes up. Describe the sample space.
Solution: In the experiment head may come up on the first toss, or the 2nd toss, or the 3rd toss and so on till head is obtained. Hence, the desired sample space is
\(
S =\{ H , TH , TTH , TTTH , \text { TTTTH, } \ldots\}
\)
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