Probability

Probability

Introduction
Many of the decisions in daily life involve probability considerations, and some times, people are even not aware that they are taking them into account. This is reflected well in language, with phrases like ‘it might rain this afternoon’, ‘l have a good feeling about this’, ‘l might get lucky today’, ‘this sounds good’, ‘this might be our day, and so many others. But as experience suggests, things might not always work out as expected. However, if you take a bird’s eye view of the world, probabilities are everywhere; for example, in weather forecasting, sports, the banking sector, insurance industry, and many other areas,
In this topic, you will understand and apply probability to solve a wide range of
problems.
11.1 Understanding the Terms; Random, Experiment, Outcome, Sample Space, Event and Probability
Shortly before a football match starts, a referee is seen tossing a coin. The referee’s action is called a trial or an experiment, since it has a well-defined set of possible results. The possible results are more than one and even if such an experiment is repeated in the same way every time, it can give different results. Hence, the experiment is random. The possible results of such an experiment are called outcomes and the set of all the possible outcomes is called a sample space. Subsets of the sample space are called
events and the ratio of the number of elements in an event to the total number of elements in the sample space is referred to as probability.
Activity 11.1 (a) (Work in groups)

Using the internet, library resources or other sources, find out the meanings of following terms.
Probability experiment
i)
iv) Equally likely events
ii) Random probability experiment
v) Biased events
iii) Probability space
Display your findings to the plenary.
Activity 11.1 (b) (Work in groups)
(a) Discuss the likelihood ofthe events given using the following words; Impossible,
even chances, certain.
i) Christmas day will be on 25th day of December this year.
ii) Tomorrow will be Friday.
iii) It will rain today.
iv) There is an animal which is both a horse and a donkey.

Exercise 1 1.1
Construct the sample spaces for the following trials.
(a) Tossing a coin once
(b) Tossing a die once
(c) Forming 2-digit numbers from the digits 2, 4 and 5 without repetition
(d) Tossing a tetrahedron, numbered from 1 to 4, once
(e) Picking a card from 20 cards numbered from 1 to 20
11.2 Constructing a Probability Space
Defining a probability space involves writing the sample space, event spaces, and the
corresponding probabilities.
Activity 11.2 (Work in groups)
(a) What are the outcomes when two coins are tossed simultaneously?
(b) What is the probability space of obtaining heads on top when two coins are
tossed simultaneously?
(c) Using the probability space oftossing a die once, record the possible outcomes
when two dice are tossed.

11.3 Determining Probabilities from Experiments and real-life data
You already know that probability is the ratio of the number of elements in an event space to the total number of elements in the sample space.
Activity 11.3 (Work in groups)
(a) Pick two dice.
(b) Throw the two dice simultaneously and record the sum of the numbers that show up.
(c) What is the probability of obtaining a sum that is even.

Probability
4) If you have one paper note of each of the denominators UGX 1,000, UGX
2,000, UGX 5,000, and UGX 10,000 in your wallet and you select a paper note
at random to put into a charity collection box, find the probability that you;
(a) give more than UGX 2,000
(b) have less than UGX 8,000 left in your wallet
(c) have more than UGX 3,000 left in your wallet
(d) give more than one sixth of the total money in your wallet
(e) give at least 18% of the total möney in your wallet
11.4 Differentiating Between Theoretical and Experimental
Probabilities
When the probability of an event is determined basing on what is expected, such
a probability is called theoretical probability. However, when the probability of
an event is determined basing on the results of a trial, such a probability is called
experimental probability.
Activity 11.4(a) (Work in groups)
(a) Get a coin.
(b) Mark one side of the coin as head (H) and the other as tail (T),
(c) Toss the coin 10 times while recording the outcomes each time.
(d) What is the probability of a head appearing.
(e) What is the probability ofa tail appearing.

Activity 1 1.4(b) (Work in groups)2.1
(a) A coin is tossed. What is the probability of a;
(ii) tail appearing
(i) head (H) appearing
Activity 1 1.4(b) (Work in groups)
(a) What is the difference between Activity 77.4(a) and Activity 77.4(b)
11.5 Identifying and understanding Mutually Exclusive and Independent Events
Identifying and understanding mutually exclusive events
Activity 11.5(a) (Work in groups)
(a) Identify an integer that is both even and odd.
(b) State the probability of getting the number you have mentioned in (a) above.
(c) Explain your findings to the class.

Exercise 11.4
1) A card is chosen at random from a pack of 52 playing cards. What is the probability that it is either;
(a) a 6 or a King
(b) a club or an ace
black or hearts
(c)
2) Adie is numbered at b, c, d, e, f. When the die is thrown, what is the probability
that the letter is a;
(b) consonant
(a) vowel
(c) letter in the word “MATHEMATICS”
(d) letter in the word “MATHEMATICS” or a vowel
3) A card is chosen at random from twenty-five cards numbered 1, 2, 3, .
, 25.
What is the probability that the number on it is a multiple of 3 or a factor of
4) Abag contains 3 blue, 4 red, and 5 green beads. A bead is selected at random
from the bag. Find the probability that the bead is;
(a) red
(b) either blue or green
(c) not red

Identifying and understanding independent events
Activity 11.5(c) (Work in groups)
(a) Identify real-life examples of events whose occurrences do not influence the occurrence of other events.
(b) If A and B are independent events, write a note about the probability of both
A and B occurring together [P(A and B)].
(c) Kiprop has 2 red and 4 blue pens in his pocket, and 3 red and 2 blue pens in his bag. A pen is picked by Chebet at random from Kiprop’s pocket and then from the bag. Find the probability that both pens are red.
(d) Compare your answers with other groups.
Exercise 11.5
1)Find the probability of getting a head and an odd number when a fair coin
and an unbiased die are tossed once at the same time.
2) A card is chosen at random from 15 cards numbered from 1 to 15. Find the probability that it bears either a multiple of 3 or a multiple of 4.

Using probability trees to determine the probabilities of independent events
Activity 11.6(c) (Work in groups)
(a) Study the following probability tree.

Exercise 1 1.6
1) a) Abooki’s box contains 3 red and 4 blue pens. He picked a pen from his box.
What is the probability that he picked;
ii) a blue pen
i) a red pen
b) As he was closing his box, his friend requested to be lent a pen. He picked another
pen from the box without looking into it. What is the probability that he picked;
i) a red pen, given that he had picked a blue pen for himself
ii) a red pen, given that he had picked a red pen for himself
iii) a blue pen, given that he had picked a blue pen for himself
iv)a blue pen, given that he had picked a red pen for himself
(2)A jar contains 4 red and 6 blue marbles. Two marbles are picked out, one after the other.
(a) Find the probability of picking a blue marble at the second picking;
i) with replacement
ii) without replacement
(b) What is the probability that the second marble picked is a red one, given
that the first one picked was blue?
(3)A bag contains 4 white and 5 black beads. A bead is picked without
replacement. What is the probability that on picking a second bead, it is;
(a) white
b) black
(4)A bag contains 3 blue, 6 green and 4 red pens. Two pens are picked at random without replacement.
(a) Construct a probability tree diagram for the experiment.
(b) Using the probability tree diagram, find the probability that;
i) a red pen is picked second, given that a green pen was picked first
ii) the second pen picked is blue
iii) the second pen picked is green
iv) if the first pen picked was red, the second is also red

5)A box contains 3 blue and 4 white balls. Three balls are picked at random without replacement.
(a) Construct a probability tree diagram for the experiment.
(b) Find the probability of picking;
i) blue or white balls but not balls of both colours
ii) balls of different colours
iii) more white than blue balls
11.7 Using Venn diagrams to Determine Probabilities
n Senior Two, you studied about Venn diagrams and how to represent information
n them. Here, you will learn how to use Venn diagrams to find the probabilities of
ccurrences of events.
Activity 11.7 (Work in groups)
(a) Carry out a survey in your class to find out the most liked co-curricular activity
by your classmates, from the following: football, athletics, and netball
(b) Represent your findings on a Venn diagram.
(c) Determine the probability that a learner selected at random likes;
v) football and netball
none of the co-curricular activities
ii) football only
iii) netball only
iv) football and netball only
vi) athletics and netball only
vii) all the co-curricular activities
(d) Find the sum of all the probabilities shown on the Venn diagram.

Exercise 11.7
(1)In a box, there are 4 black and 5 red balls which are identical. A ball is picked at random from the box and then replaced before another ball is picked. Find
the probability that;
(a) the first ball is black and the second ball is red
(b) both balls picked are red
(c) the balls are of the same colour
(2)A basket contains 3 oranges, 4 apples and 2 lemons. Two fruits are selected at random;
(a) with replacement
(b) without replacement
(3)Draw probability tree diagrams to show the possible outcomes. Find the probability, in each case, that the two fruits selected are of;
i) the same type
ii) different types
(4)In a family of 5 children, 2 are boys. Two children are chosen at random. What is the probability that the two children chosen are;
(a) boys
(b) girls
(c) a boy and a girl

4) At a graduation party, three types of sauce were served: meat (M),
chicken (C), and groundnuts (G)
50 guests were served with meat.
52 guests were served with chicken.
44 guests were served with groundnuts.
30 guests were served with meat and chicken.
26 guests were served with meat and groundnuts.
27 guests were served with chicken and groundnuts.
The number of guests at the graduation party was 76.
Find the probability that a guest chosen at random was served with;
(b) exactly two types of sauce
(a) all the three types of sauce
(5)A die was thrown once. What was the probability of getting;
(b) a number less than 4
(a) a prime number
(c) an even number or a number less than 4
6) Two dice are thrown together and their scores recorded as ordered pairs; for
example, (1, 1) and (1, 2). Find the probability that;
(a) the dice show the same numbers
(b) both dice show odd numbers
(c) the dice show products of their scores as 20
(d) chicken only

ICT Activity
Design a spinning wheel which will be used to award prizes in the forthcoming house competition events. Each colour has 4 pieces. The colours are green, red, blue, and yellow.

Revision Questions:
l) A fair die is rolled three times and the sums of the scores it shows on top are recorded. Determine the probability that the sum of the scores on the three rolls is less than 5.
2) Omondi is given a standard deck of cards. What is the probability that he draws three hearts, one after the other, if he draws the cards without replacement? 3) Two oranges are randomly selected from a basket containing 14 ripe oranges and 11 raw oranges. Find the probability that the selection consists
Of a ripe orange and a raw one, in any order.
4) In a box, there are 4 green balls, 2 black balls, and 1 red ball. Two balls are chosen, with replacement, from the box. By drawing a probability tree diagram and assuming that all the balls are equally likely to be chosen, find the probability that;
(a) exactly one black ball is chosen
(b) at most one black ball is chosen
(c) two balls of different colours are chosen

5)In a survey done at a school, 50% of the learners owned a calculator, owned a geometry set and 15% owned both. If there were 400 learners altogether in the survey, how many of then owned neither a calculator nor a geometry set?