Probability

ELATE: E-LEARNING AND TEACHER EDUCATION

TEACHERS’ GUIDE

Subject: Mathematics

Unit No: 4

Target group: Senior 3

TOPIC: PROBABILITY

Probability theory is the branch of Mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables and events. Although an individual coin toss or roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative Mathematics results describing such patterns are the law of large numbers.

As a Mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex system given only partial knowledge of their state, as in statistical mechanics.

The Mathematics theory of probability has its roots in attempts to analyze games of change by Gerolamo Cardano in the 16^th Century.

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical consideration compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, the foundation of which was variables into the theory.

Sub-topic: Theoretical Probability

Time Required: Minimum 80 minutes. Maximum 120 minutes

Whenever an event is carried out like tossing a coin, there is always an outcome. For example when a coin is tossed once, either a side with “Tail” or “Head” appears on top. Each of these two possibilities is called an outcome when a die is rolled once, there are six possible outcomes, a score of {1, 2, 3, 4, 5, 6}. If these scores have equal chances of appearing, then the die is said to be fair.

Probability (P) is defined as,

‘the number of ways the events can occur’

the number of possible outcomes

For example: A fair die is rolled once. Calculate probability of getting

1. an even number
2. a prime or an even number
3. a score of 6
4. a score of 8
5. scores 1, 2, 3, 4, 5, 6.

Solution:

Possible outcomes is {1, 2, 3, 4, 5, 6}

1. P(even number)

Even number = {2, 4, 6}

P (even number) = ³/₆ = ½

2. Set of a Prime or an Odd number is {1, 2, 3, 5}

P (Prime or an Odd number) = ⁴/₆ = ²/₃

3. A score of 6 is {6}

P (a score of 6) = ¹/₆

4. P(a score of 8) = ⁰/₆ = 0

5. P(score of 1, 2, 3, 4, 5, 6) = ⁶/₆ = 1

The above example demonstrates that a probability of any event (A) is always

0 ≤ P(A) ≥1. 0 ≤ P(A) ≥1.

The probability of an event equals one if the occurancy of that event is “certain”. For example, if we roll a die what is the probability of getting a score less than 7? There are 6 possible outcomes: 1, 2, 3, 4, 5, 6.

All these outcomes are scores less than 7 so P(a score less than 7) = ⁶/₆ = 1. What is P (a score of 7)?

No outcome gives a score of 7, so P (7) = 0.

The probability of a certain event is 1.

The probability of an impossible event is 0.

Exercise

1. Nine pieces of paper are put in a bag. One piece has a 1 on it, one piece has a 2 on it and so on up to 9. one piece of paper is picked at random from the bag. Find the probability of these events.

1. The number on the paper is ‘odd’.
2. The number on the paper is either ‘prime’ or a ‘multiple of 2’.
3. The number is less than ‘8’ and a multiple of ‘2 and 3’.

2. A number is selected at random from the sets of integers from 1 to 20 inclusive. Find the probability that the number selected is.

1. a prime number
2. an odd number
3. a multiple of 2
4. a multiple of 3
5. a multiple of 5
6. a multiple of both 2 and 3
7. a multiple of 5 or 7
8. the square of 4
9. the square of 5.

Sub-topic: Experimental Probability

It is possible to calculate the probability of any event. In many cases, it may not be possible to calculate the probability in this case probability can be determined or estimated through experiments.

Example: What is the probability that if we select a football team in Uganda that has scored less than 2 goals on a given week end?

Number of goals	0	1	2	3	4	5
Number of teams	4	5	4	2	3	2

There are 4 + 5 + 4 + 2 + 3 + 2 = 20 teams

Teams which scored less than two goals are 4 + 5 = 9.

So the experimental probability that the team has scored less than 2 goals = ⁹/₂₀.

Note:

In order to get the scores of each team that played on that week end, you need to watch the matches or get the results from other sources e.g. newspapers, football fans, electronic media etc.

In theory, when a fair coin is tossed twice, we expect to get one head or one tail, if it is tossed 10 times then results would be 5 heads or 5 tails.

If the tossing is done practically / experimentally the results may not be the same as those obtained theoretically. So if a coin is tossed 50 times, the number of times a head or tail appears may not be the same. For instance you may get 29 heads and 21 tails but if the coin is tossed several times like 100 times, the number of times the head appears is approximately equal to the number of times the tail appears.

Insert an experiment of determining the number of times a head appears when a coin is tossed

1. 20
2. 40
3. 50
4. 70
5. 100
6. 150,

(using a computer)

Exercise

1. In a traffic survey, the number of passengers (including the driver) in each car passing a school was recorded and the results were as follows:

Number of passengers	1	2	3	4	5	6	7	8
Number of cars	15	20	12	19	14	10	6	5

What is the probability that a car picked at random has;

1. exactly 4 passengers?
2. more than 2 passengers?
3. less than the average number of passengers?

2. In a survey, the heights of students of Senior 3 were recorded and results were as follows:

Height of pupils	140-149	150-159	160-169	170-179	180-189
Number of pupils	2	12	14	21	16

a) What is the probability that a student picked at random is;

(i) more than 169 cm tall?

(ii) less than 160 cm tall?

b) The basket ball coach wants to pick a team. He says that all players must

be at least 160 cm tall. What is the probability that a student chosen at

random will meet the coach’s requirements?

Sub-topic: Possibility space in Cartesian Diagrams

Time Required: Minimum 80 minutes. Maximum 120 minutes

All the possible outcomes when a coin is tossed or a die is rolled or when a coin and a die are tossed form what is called possibility space.

For example:

1. When a coin is tossed once the possibility space is {H, T}.

2. A die is rolled, the possibility space is {1, 2, 3, 4, 5, 6}.

3. 2 dice are rolled, the possibility space is

(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, 3), (6, 5), (6, 6)

4. A die and a coin tossed, the possibility space is

(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)

(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6).

Note:

• Explain what each of the pairs in (c) and (d) means e.g. (4, 2), (T, 4), (4, 2) means that the first die shows up the side marked 4 and the second dies shows 2.
• (T, 4) means that a coin is tossed first and shows the “Tail” and the die shows up 4.
• (c) and (d) above can be put in Cartesian diagram

Second die	First die
		1	2	3	4	5	6
	1	1, 1	1, 2	1, 3	1, 4	1, 5	1, 6
	2	2, 1	2, 2	2, 3	2, 4	2, 5	2, 6
	3	3, 1	3, 2	3, 3	3, 4	3, 5	3, 6
	4	4, 1	4, 2	4, 3	4, 4	4, 5	4, 6
	5	5, 1	5, 2	5, 3	5, 4	5, 5	5, 6
	6	6, 1	6, 2	6, 4	6, 4	6, 5	6, 6

Example:

Use the possibility space for throwing two dice to calculate these probabilities

1. P (a sum of 10 or more)
2. P (an even sun)
3. P (a sum of less than 12)
4. P (a sum of 13).

Solution:

Second die	First die
		1	2	3	4	5	6
	1	2	3	4	5	6	7
	2	3	4	5	6	7	8
	3	4	5	6	7	8	9
	4	5	6	7	8	9	10
	5	6	7	8	9	10	11
	6	7	8	9	10	11	12

The possibility space is 36

1. P (a sum of 10 or more) = ⁶/₃₆ = ¹/₆
2. P (an even sum) = ¹⁸/₃₆ = ½
3. P (a sum of less than 12) = ³⁵/₃₆
4. P (a sum of 13) = ⁰/₃₆

Exercise

1. A number is selected at random from the set S = {1, 2, 3, …, 18}

1. Find P (prime number).
2. Find P (even number).
3. Find P (multiple of 6).

2. Construct the possibility space for tossing 4 coins and use it to calculate these probabilities.

1. P (4 heads).
2. P (less than 4 heads).
3. P (2 heads).

Sub-topic: Independent or Mutually exclusive events

Time Required: Minimum 80 minutes. Maximum 120 minutes

One of the important steps you need to make when considering the probability of two or more events occurring. Is to decide whether they are independent or related events.

For example:

Mutually Excusive Vs Independent

It is not uncommon for people to confuse the concepts of mutually exclusive events and independent events.

Definition of a mutually exclusive event

If event A happens, then event B cannot, or vice-versa. The two events “it rained on Tuesday” and “It did not rain on Tuesday” are mutually exclusive events. When calculating the probabilities for exclusive events you add the probabilities.

Independent events

The outcome of event A, has no effect on the outcome of event B. Such as “It rained on Tuesday” and “My chair broke at work”. When calculating the probabilities for independent events you multiply the probabilities. You are effectively saying what is the chance of both events happening bearing in mind that the two were unrelated.

To be or not be …?

So, if A and B are mutually exclusive, they cannot be independent. If A and B are independent, they cannot be mutually exclusive. However, if the events were “it rained today” and “I left my umbrella at home” they are not mutually exclusive, but they are probably not independent either, because one would think that you would be less likely to leave your umbrella at home on days when it rains. That fact aside use the following to understand the definition.

Example of a mutually exclusive event

What happens if we want to throw 1 and 6 in any order? This now means that we do not mind if the first die is either 1 or 6, as we are still in with a chance. But with the first die, if 1 falls uppermost, clearly it rules out the possibility of 6 being uppermost, so the two outcomes, 1 and 6 are exclusive. One result directly affects the other. In this case, the probability of throwing 1 or 6 with the first die is the sum of the two probabilities, 1/6 + 1/6 = ⅓.

The probability of the second die being favourable is still 1/6 as the second die can only be one specific number, a 6 if the first die is 1, and vice versa.

Therefore, the probability of throwing 1 and 6 in any order with two dice is ⅓ x ¹/₆ = ¹/₁₈. Note that we multiplied the last two probabilities as they were independent of each other!!!

Example of an independent event

The probability of throwing a double three with two dice is the result of throwing three with the first die and three with the second die. The total possibilities are, one from six outcomes for the first event and one from six outcomes for the second, therefore (1/6) * (1/6) = 1/36^th or 2.77%.

The two events are independent, since whatever happens to the first die cannot affect the throw of the second, the probabilities are therefore multiplied, and remain 1/36^th.

The probability of getting H and H when 2 coins are tossed {HT, HH, TH, TT} is ¼.

Note that P (H, H) = ¼ = ½ x ½ = P (H) x P (H)

By definition if A and B are independent events P ( (A, B) = P (A) x P (B)

We can also determine the probability of independent events by using Tree diagram.

When two coins are tossed, the outcomes can be illustrated in the following diagram:

First Second Outcome

toss toss

So, P (H, H) = P (H) x P (H) = ½ x ½ = ¼

P (H, T) = P (H) x P (T) = ½ x ½ = ¼

P (T, T) = P (T) x P (T) = ½ x ½ = ¼

P (T, H) = P (T) x P (H) = ½ x ½ = ¼

The above diagram is what is called Tree diagram. The above Tree diagram can be used to determine probabilities of events e.g.

P (one head) = P (H, T) + P (T, H)

= ¼ + ¼ = ½.

The above probabilities have been added together because the events (H, T) and (T, H) are mutually exclusive. This means that when one event occurs the other does not.

Example 1:

A bag contains 3 red and 9 white beads, two beads are taken out of it with replacement. Draw a tree diagram and use it to find these probabilities

1. P (two white beads are picked)
2. P (one white and one red bead picked).

Solution:

Possibility space = 3 + 9 = 12

1. P (red) = P (r) = 3/12
2. P (white) = P (w) = 9/12

1. P (w, w) = P (w) x P (w) = 9/12 x 9/12 = ¾ x ¾ = 9/16
2. P (w, r) = P (w, r) + P (r, w)

= P (w) x P (r) + ( P (r) x P (w)

= ⁹/₁₂ x ³/₁₂ + ³/₁₂ x ⁹ /₁₂

= ¾ x ¼ + ¼ x ¾

= ³/₁₆ + ³/₁₆ = ⁶/₁₆

Example 2:

Use the above example without replacement.

Solution:

1^st pick, the possibility space is 3 + 9 = 12

P (w) = 9/12

P (r) = 3/12

2^nd pick, the possibility space is 12 -1 = 11

P (w) depends on whether, in first pick the bead was white.

If it was white, then P (w) = 8/11

If it was not, P (w) = 9/11, likewise

If it was red then P (r) = 2/11

If it was not red, P (r) = 3/11

Diagrammatically, it be illustrated as follows.

⁸/₁₁ w

w

⁹/₁₂ ³/₁₁ r

© ³/₁₂ ²/₁₁ r

r

⁹/₁₁ w

So,

1. P (w, w) = P (w) x P(w) = ⁹/₁₂ x ⁸/₁₁ = ⁷²/₁₃₂ = ¹⁸/₃₃
2. P (w, r) = P (w r) + P(r w)

= P (w) x P(r) + (P (r) x P(w)

= ⁹/₁₂ x ³/₁₁ + ³/₁₂ x ⁹ /₁₁

= ¾ x ¼ + ¼ x ¾

= ⁹/₄₄ + ⁹/₄₄ = ¹⁸/₄₄ = ⁹/₂₂

Exercise

1. A basket contains 6 mangoes and 4 oranges. Two fruits are removed from it without replacement. Use tree diagrams to work out the following probabilities.

1. P (three mangoes are removed)
2. P (a mango and two oranges are removed)

2. A bag contains 5 blue pens and 3 red pens, three pens are removed with replacement. Find the probability that;

1. All the pens are of the same colour
2. The pens are not all the same colour
3. More blue than red pens are picked.

Sub-topic: Probability from simple Venn diagrams

Time Required: Minimum 80 minutes. Maximum 120 minutes

Venn diagrams which are usually used in set theory can also be used to solve some probability problems.

Example 1:

The diagram shows how children come to school by walking (W), by bicycle (B) or by car (C).

Use the information on the Venn diagram to find the probability that a child picked at random

1. walks
2. uses a car
3. uses a bicycle
4. walks only
5. uses a bicycle and car
6. uses all the means of travel.

Solutions:

Possibility space = 6 + 5 + 7 + 10 + 9 + 8 + 12 = 57

1. children who walk = 7 + 8 + 5 + 9 = 29

P (W) = ²⁹/₅₇

2. uses a car = 6 + 5 + 10 + 9 = 30

P (C) = ³⁰/₅₇ = ¹⁰/₁₉

3. uses a Bicycle = 10 + 9 + 8 + 12 = 39

P (B) = ³⁹/₅₇ = ¹³/₁₉

4. Walks only = 7

P (child who walks only)= ⁷/₅₇

5. Uses Bicycles and Car = 10 + 9 + 8 = 27

P (child who uses Bicycle and Car) = ²⁷/₅₇ = ⁹/₁₉

6. Uses all the means of travel = 9

P (Child uses all the means of travel) = ⁹/₅₇ = ³/₁₉

Example 2:

In a class, 15 pupils like nature study, 13 like crafts and eight like both subjects. Six pupils do not like either subject. What is the probability that a pupil picked at random from the class does not like crafts?

Solution:

Crafts (C), Nature Study (N)

Pupils who don’t like crafts = pupils who like Nature Study only + pupils who don’t like any of the subjects = 7 + 6 = 13.

Possibility space: 7 + 8 + 5 + 6 = 26

P (Pupils picked at random does not like Craft) = ¹³/₂₆ = ½

Exercise

1. A and B are two sets of numbers such that ∩(a) = 17, ∩(B) = 10 and ∩(AUB) = 25. Use a Venn diagram to find the probability that a number picked at random from AUB is a member of A∩B.

2. In a car park, there were 22 cars. Ben noticed that 12 of them were blue. He also noticed that nine had sun roofs. Only two blue cars had sun roofs. Show this information on a Venn diagram. What is the probability that a car chosen at random had not sun roof?

3. AUB consists of the whole numbers from 1 to 29, so ∩(AUB) = 29, ∩(A) = 17 and ∩(B) = 23. Find the probability that;

1. a number chosen at random from AUB is in A but not in B.
2. a number chosen at random from A is not in B.
3. a number chosen at random from A is not in AUB

LESSON PLAN

Date	Class	Period	Number of students
…	Senior 3	1 – 2	…

Topic: Probability

Sub-topic: Experimental Probability

Lesson Objectives:

By the end of this lesson, the students should be able to:

1. differentiate theoretical from Experimental probability
2. calculate probabilities from Experiments.

Methods: Discussion, Group work, demonstration

Teaching aids: coins, dice, manila paper, pens/pencils

References:

Secondary school Mathematics Bk 3.

School Mathematics for East Africa Bk 3.

Mathematics for Kenya Schools Bk 3.

Steps	Time	Content	Teachers’ Activity	Students’ Activity
1	5 mins	Roll call	Reads student’s names	Answer
2	10	Review of the previous work of Theoretical probability	Asks students on the previous work	Answer the Teachers’ questions.
3	10	“	Answers questions asked by students	Seek guidance from the teacher on the content they did not understand.
4	15 mins	Demonstration on experimental probabilities	Demonstrates using coin or die	Observe what the teacher is demonstrating
5	25 mins	Experimental probability	Moves around in the class observing what the students are doing	Students carry out tossing coins and rolling dice and record the outcomes on manilla paper.
6	10 mins	Class exercise	Gives probability questions to be calculated by students	In their groups, students answer the asked questions
7	5 mins	Conclusion and exercise	Recaps the sub-topic and gives exercise	Record the exercise and attempt the exercise.

Black Board Plan

Date	Topic
	Notes	New Words

Self-Evaluation:………………………………………………………………

PROBABAILITY

In a group of six learners carry out the following tasks and give your agreed answer / opinion.

1. Go out of class and look at the sky

Question: Will it rain in the next one hour?

2. Agree amongst yourselves and tick the most appropriate answer as a group.

It must rain	Most likely	Likely	Least likely	Not sure	It cannot rain

3. Get 4 small tins and label them T1, T2, T3 and T4. Half fill them with water and cover T3 and T4 with their lids and rest them on a table.
4. If the tins are knocked to fall off the table, what is the probability that
   1. The water in open tins will pour?
   2. The water in closed tins will pour?
5. Which tins have no chances of water pouring?
6. Knock T1, then T2, T3 and T4 and record your observations in the table below:

	What happens to water
Tin 1
Tin 2
Tin 3
Tin 4

7. Do the observations match your answers in (d) above?
8. Consider Sarah who left her kitchen open, and in the neighborhood there was a wild cat. How safe is her piece of roasted meat which she lest in the kitchen? Agree on which alternative to tick.

Very safe	Safe	Possibly safe	Not safe at all

9. What precautions should Sarah have taken?
10. Mention some circumstances when one can get a problem yet it could have been avoided or reduced its changes of happening.

Possible answers (to be hidden as hyperlink)

1. Driving while drunk
2. Taking strong drugs without doctor’s prescription
3. Sailing without a life jacket