INTRODUCTION TO STATISTICS AND PROBABILITY SOLVED
PROBLEMS
,Question 1
SOLUTION
1.1) Basic properties that apply to every probability problem are;
a) For any given event say event A, the probability value will lie between zero and 1
i.e. 0 ≤ P (A) ≤ 1. Probabilities are usually converted to percentages and the chance
of any event occurring cannot be more than 1 or less than 0 (Wegner, 2013).
b) In case there is no chance for an event to happen or the occurrence of a certain
event is impossible then the probability value assigned to such an event is 0 (zero),
thus, P (A) = 0 (Anderson, et al., 2011).
c) In cases where there is 100% certainty that an event A will occur, then a probability
value of 1 (one) is assigned to the event, thus, P (A) = 1 (Wegner, 2013).
d) The sum of probabilities of all events that make up a sample space is equal to one.
That is if an event is made up of two possible outcomes then P (A) + P (B) =1. An
example would be the outcomes of the experiment, tossing a fair sided coin. If a
coin is tossed, there are only two possible outcomes, which is either obtaining a
head or a tail each with a probability value of 0.5 and the sum gives one. Thus, is
there are n possible events in an experiment then P (A1) + P (A2) + P (A3) + …+ P
(An) =1 (Keller, 2014).
, e) The complement rule – If the probability of an event A happening is given by P(A),
then the probability that this even will not occur under any circumstances is given
by P(Ac) = 1 – P(A) (Wegner, 2013).
1.2.1 Collectively exhaustive events – These are events whose union make up the
sample space. That is, if all the possible events in a sample space are put together the
sample space is produced (Wegner, 2013).
1.2.2 Mutually exclusive events are events are cannot occur at the same time. These are
referred to as disjoint events, If two events A and B are mutually exclusive, then their joint
probability is equal to zero i.e. P (A and B) = 0 (Keller, 2014).
1.2.3. Two events A and B are said to be statistically independent if the occurrence of
one event A does not have an influence on the occurrence of the other event B. In simple
terms, the outcome of event A does not affect the outcome of event B (Wegner, 2013).
1.3. Venn Diagram
A B
A and B
Sample Space
PROBLEMS
,Question 1
SOLUTION
1.1) Basic properties that apply to every probability problem are;
a) For any given event say event A, the probability value will lie between zero and 1
i.e. 0 ≤ P (A) ≤ 1. Probabilities are usually converted to percentages and the chance
of any event occurring cannot be more than 1 or less than 0 (Wegner, 2013).
b) In case there is no chance for an event to happen or the occurrence of a certain
event is impossible then the probability value assigned to such an event is 0 (zero),
thus, P (A) = 0 (Anderson, et al., 2011).
c) In cases where there is 100% certainty that an event A will occur, then a probability
value of 1 (one) is assigned to the event, thus, P (A) = 1 (Wegner, 2013).
d) The sum of probabilities of all events that make up a sample space is equal to one.
That is if an event is made up of two possible outcomes then P (A) + P (B) =1. An
example would be the outcomes of the experiment, tossing a fair sided coin. If a
coin is tossed, there are only two possible outcomes, which is either obtaining a
head or a tail each with a probability value of 0.5 and the sum gives one. Thus, is
there are n possible events in an experiment then P (A1) + P (A2) + P (A3) + …+ P
(An) =1 (Keller, 2014).
, e) The complement rule – If the probability of an event A happening is given by P(A),
then the probability that this even will not occur under any circumstances is given
by P(Ac) = 1 – P(A) (Wegner, 2013).
1.2.1 Collectively exhaustive events – These are events whose union make up the
sample space. That is, if all the possible events in a sample space are put together the
sample space is produced (Wegner, 2013).
1.2.2 Mutually exclusive events are events are cannot occur at the same time. These are
referred to as disjoint events, If two events A and B are mutually exclusive, then their joint
probability is equal to zero i.e. P (A and B) = 0 (Keller, 2014).
1.2.3. Two events A and B are said to be statistically independent if the occurrence of
one event A does not have an influence on the occurrence of the other event B. In simple
terms, the outcome of event A does not affect the outcome of event B (Wegner, 2013).
1.3. Venn Diagram
A B
A and B
Sample Space
