Question

Practice.
1. Two coins are tossed. Give examples of
a) Sure event;
b) Impossible event;
c) Random event;
d) Elementary random event.
2. In 3 tosses of a coin which of following equals the event “exactly two heads”?
a) ? = {???, ???, ???, ???};
b) ? = {???, ???, ???};
c) ? = {???, ???}.
3. Give an answer of
a) ? ∪ ?̅ =… ;
b) ? ∩ ?̅ =… .

Theory. Explain the concept of random or stochastic experiment, elementary random event, set of
elementary events (sample space), random or stochastic event, impossible and certain (or sure) event, Venn
diagrams.

Answer 1

1.

a)  A sure event

A sure event is the one that contains the whole sample space.

C=" get head in one of the two coins, or get the same result in both "

b) Impossible event:

An impossible event is the opposite case, when the event does not contain any element of the sample space.

D=" neither get head in one of the two coins, or neither get the same result in both "

Ω’ is an impossible event.

c) random event:

The events whose outcome is unknown are called random experiments. For example when we toss two coin, we do not know if it will land get head in one of the two coins, or get the same result in both. Hence tossing two coins simultaneously is a random experiment.

Examples getting two heads, getting two heads,getting one tail, getting at least 1 head etc.

d) Elementary random events:

an elementary random event  is an event which contains only a single outcome in the sample space.

if a coin is tossed twice then {HH}, {HT}, {TH} and {TT} is elementary random event

2. .

Event B and C  are event having “exactly two heads”.

Because event A have " three heads "

3.

a) Ω (sample space)

b) Ω’ ( null )

THEORY

Random or stochastic experiment:

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables A stochastic process is defined as a collection of random variables X={Xt:t∈T} defined on a common probability space, taking values in a common set S (the state space), and indexed by a set T, often either N or [0, ∞) and thought of as time (discrete or continuous respectively)

elementary random event:

The events that are influenced by chance are called random events.To study chance and its properties we use random experiments, for example: to roll a die, to flip a coin etc.

Elementary Events are unique events in the sample space.

set of elementary events (sample space):

the sample space  of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U . The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite

Sample Space are the set of events that are possible.

A random or stochastic event

A random event is something unpredictable. As it is unpredictable, you can never give it an exact value / probability. For example, you couldn't predict the probability of you falling down a flight of stairs in the next ten years, as that's a completely random event

Impossible and certain (or sure) event:

An impossible event is the opposite case of sure event, when the event does not contain any element of the sample space.

certain event:

A certain event is certain to occur. If event A is certain, then P(A) = 1.

Venn Diagram:

A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Often, they serve to graphically organize things, highlighting how the items are similar and different.