The document provides guidance on teaching a lesson on probability. It outlines key concepts to define, such as experimental and theoretical probabilities. It also describes solving problems involving combined probabilities of mutually exclusive and independent events. Example problems are provided for different probability concepts. The teacher is instructed to check students' understanding through oral questions and exercises.

2. By the end of the lesson, the students should be able to;
≈ Define Probability.
≈ Use the language of probability including applications to set
language, to describe and evaluate events involving chance.
≈ Define and use experimental probabilities to estimate
problems involving chance.
≈ Define and use theoretical probabilities to calculate
problems involving chance.
≈ Solve problems involving combined probabilities particularly
those relating to mutually exclusive events and independent
events.
≈ Solve problems on probability experiments with or without
replacement.

3. Steps Content
Development
Teacher’s Activities Students'
Activities
Instructional
Materials
Instructional
Techniques
1 Probability The teacher defines probability
as the mathematical expression
of the extent to which a n event
is likely to occur, given a value
between 0(an impossibility)
and 1 (a certainty). And
explains some terms used in
probability like experiment,
outcome, events etc.
He goes ahead to explain and
use experimental probabilities
to estimate problems involving
chance.
He equally explains and uses
theoretical probabilities to
calculate problems involving
chance.
He then solves problems
involving combined
probabilities particularly those
relating to mutually exclusive
events and independent events.
The students answer
questions like the
total number of
outcomes in a fair
coin, dice, pack of
playing cards orally
and ask questions
where they are
confused. And then
solve questions
picked from the
exercises in their
everyday
mathematics.
1. A ludo
game.
2. Some dice.
3. Some
coins.
4. Pack of
playing
cards.
5. Addition
rule chart.
Etc
Demonstr
ation,
Illustratio
n,
Repetition
,
Questioni
ng and
answering
.

4. In order to ascertain the extent to which the
students understood the lesson, the teacher
asks the students oral questions like the
total number of outcomes in a coin, a fair
die, the probability of getting a 4 in a ludo
game of two dice etc, and finally giving them
some exercises to do in their everyday
mathematics.

5. In our day-to-day activities in life, our expectations
may either come to pass or not. For instance, a
trader’s expectation in his business is to make profit,
but, sometimes, it does not happen as expected.
Whatever happens, everything we attempt to do in life
is a trial, while the result of such an attempt is the
outcome.
The process of tossing a coin or throwing a die is an
experiment while the results are the outcomes. In
tossing a coin once, the result can either be for the
coin to show a head or a tail. Therefore, there is a total
of two outcomes. If it is a head, the chance is one out
of the total outcomes, i.e. ½. The ½ (which is the
chance of getting a head) is therefore known as the
Probability of getting a head.

6. Probability is therefore defined as the mathematical expression
of the extent to which an event is likely to occur, given a value
between 0 (an impossibility) and 1 (a certainty).
The probability of an event x happening is the ratio of the
required outcome to the total number of possible outcomes.

7. What is the probability of getting a 3 on throwing a
fair die once?
Solution
Total number of possible outcomes = 6
Number of required outcomes (i.e. number of times 3
appeared on a fair die) = 1

8. Out of 200 apples bought from a fruit seller, it is known
from experience that 10 will be unripe. What is the
probability that the apple picked will be ripe?
Solution
Total number of possible outcomes = 200
Number of unripe apples = 10
Therefore number of ripe apples = 200 – 10
= 190

9. Since experimental probability cannot be taken to be perfectly accurate as it is
based on numerical data of past experiences to predict the future, therefore,
determining probability has been further clarified with the introduction of
the theoretical concept.
Theoretical probability bases its results and occurrences on exact values that
are dependent on the physical nature of the situation under consideration.
For instance, if the probability of an event happening is p, the p lies between
0 and 1,
ie. 0 < p < 1, but the probability that an event is not happening is 1 – p. Then
it follows that the sum of an event happening and event not happening is
always equal to one, (1), i.e. p + (1-p) = p + 1 – p = p – p + 1 = 1.
Probability can also be denoted in set language. If the probability of an event
happening is pr (R),

10. If a pair of fair dice are thrown once, what is the probability that :
A sum of 5 is obtained
A sum of 7 is obtained
A sum of an event number greater than 2 but less than 12 is obtained
Solution
Sum of 5 can occur 4 ways and there are 36 possible outcomes.
Therefore Pr (sum of 5) = n(R) / n(U)
But n(R) = 4 and n(U) = 36
Hence, Pr (sum of 5) = 4/36 = 1/9
Sum of 7 can occur 6 ways.
Therefore. Pr (sum of 7) = 6/36 = 1/6
n (even number greater than 2 but less than 12) = 13
Therefore Pr (sum of 2 <£ < 12) = n(R) / n(U) = 13/36

11. When an event prevents the occurrence of another, we say that the
two events are mutually exclusive because the two events cannot
happen at the same time. For instance, it is impossible to throw a
die and score a three and a four at the same throw with a single
die. The task of
scoring either a three or a four involves separate events which
cannot happen together, ie. one event excludes the other. The
separate probabilities are added together to give a combined
probability.
Mutually exclusive events can be referred to as addition of
probability. If X and Y are mutually exclusive, Pr (X or Y) = Pr (X)
+ Pr (Y), ie. X and Y cannot occur at the same time.
It is important to note that the events that are not mutually
exclusive are events that occur or happen at the same time. The
probability that both will occur is equal to the product of their
probabilities.

12. In a basket, there are 8 oranges, 7 pawpaw and 5 water melons. What
is the probability of picking
An orange
A pawpaw
An orange or a pawpaw
Solution
Total Possible outcomes = 8+7+5 = 20
Hence probability of picking an orange = 8/20 = 2/5
Probability of picking a pawpaw = 7/20
Probability of picking an orange or a pawpaw = Pr (an orange or a
pawpaw) = 2/5 + 7/20 = 15/20 = ¾

13. If two events X and Y can occur without affecting each
other, then the two events X and Y are said to be
independent events.
The probability of two or more independent events is
found by multiplying the individual probability of each of
the events that are involved. To identify independent
events, words like ‘and', ‘both' and ‘all 'are usually used.
Independent events in probability are also referred to as
multiplication of probability. In case of independent events,
the probabilities of all the events are multiplied to give the
combined probability.

14. If events X, Y, Z,…, are independent, the probability of
X and Y and Z… happening is the product of the
probabilities of all of them, i.e. Pr (X) x Pr (Y) x Pr (Z)
x …

15. Three balls are drawn one after the other with
replacement, from a bag containing 3 red, 6 white and 3
blue identical balls. What is the probability that they are
one red, one white and one blue?
Solution
Total possible outcomes = 3 + 6 + 3 = 12
Pr (one red) = 3/12 = ¼
Pr (one white) = 6/12 = ½
Pr (one blue) = 3/12 =1/4
Therefore Pr (one red, one white, one blue)
= ¼ * ½ * ¼ = 3/12