Mathematics PROBABILITY - Introduction , Random Experiments and Outcomes and sample space
Click for Only Video

### Topics Covered

star Introduction
star Random Experiments
star Outcomes and sample space

### Introduction

\color{red} ✍️ We have obtained the probability of getting an even number in throwing a die as 3 / 6 i.e., 1 /2 .

Here the total possible outcomes are 1,2,3,4,5 and 6 (six in number). The outcomes in favour of the event of ‘getting an even number’ are 2,4,6 (i.e., three in number).

\color{red} ✍️ In general, to obtain the probability of an event, we find the ratio of the number of outcomes favourable to the event, to the total number of equally likely outcomes. This theory of probability is known as color(blue)"classical theory of probability."

### Random Experiments

\color{red} ✍️ We also perform many experimental activities, where the result may not be same, when they are repeated under identical conditions.

color(red)("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 color(navy)"random experiments. "

color(green)(ul"An experiment is called random experiment if it satisfies the following two conditions :")

color(red)((i)) It has more than one possible outcome.

color(red)((ii)) It is not possible to predict the outcome in advance.

### Outcomes and sample space

\color{fuchsia} {ul ★ "Outcome"}

A possible result of a random experiment is color(green)("called its outcome.")

color(red)(=>"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 color(navy)"sample space of the experiment."

\color{fuchsia} {ul ★ "Sample space "}

Thus, the set of all possible outcomes of a random experiment is color(green)("called the sample space") associated with the experiment.

color(green)("Sample space is denoted by the symbol") color(navy)(S.)

\color{fuchsia} {ul ★ "Sample point "}

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."
Q 3130891712

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}
Q 3150891714

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). Simlarly, 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 x× 6 = 36 and the sample space is
given below:
{(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)}
Q 3170891716

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,...}
Q 3100891718

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 = { HB_1, HB_2, HB_3, HW_1, HW_2, HW_3, HW_4,
T1, T2, T3, T4, T5, T6}.
Here HBi means head on the coin and ball Bi is drawn, HW_i means head on the coin
and ball Wi is drawn. Similarly, Ti means tail on the coin and the number i on the die.
Q 3110891719

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, T TH, T T T H, T T T T H,...}