Topic 4: Probability - Basic Mathematics Form Four

Topic 4: Probability – Basic Mathematics Form Four

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Basic Mathematics For Form Four Full Notes, LINEAR PROGRAMMING. Linear programming – is a branch of mathematics which deals with either minimizing the cost or maximizing the profit. It gives the best way of utilizing the scarce resources available. It is so called because it only involves equations and inequalities which are linear. Simultaneous Equation. One of the methods used in solving linear simultaneous equations is a graphical method. Two linear simultaneous equations in two unknowns can be graphically solved by passing through the following procedures. Draw the two lines which represent the two equations on the xy – plane this is done by deter mining at least two points through which each line passes, the intercept are commonly used Determine the point of intersection of the two lines. This point of intersection is the solution to the system of equations. FACT: If two straight lines are not parallel then they meet at only one point: In case the lines do not meet, there is no solution to the corresponding system of simultaneous equations. Example 1 Graphically solve the following system of simultaneous equations. Example 2 Find the solution to the following system of simultaneous equations by graphical method. Solving Simultaneous Equations Graphically Solve simultaneous equations graphically Example 3 Solve the following simultaneous equations graphically and check your solution by a non-graphical method: Example 4 Find the solution to the following system of simultaneous equations by graphical method. Exercise 1 Find the solution to the following systems of simultaneous equations graphically. Try: Ali paid 34 shillings for 10 oranges and 35 mangoes. Moshi went to the same market and paid 24 shillings for 16 oranges and 18 mangoes. What was the price for a mango and for an orange? Inequalities Forming Linear Inequalities in Two Unknowns from Word Problems Form linear inequalities in two unknowns from word problems Linear inequalities Normally any straight line drawn on xy – plane separates it into two disjoint sets. These sets are called half – planes Consider the equation y = 5 drawn on the xy plane as shown below. From the figure above, all points above the line, that is all points in the half plane A which is above the line satisfy the relation y>5 and those lying in the half plane B which is below the given line, satisfy the relation y< 5. Shading of Regions In linear programming usually the region of interest is left clear that is we shade unwanted region(s). NB: When shading the half planes we consider the inequalities as the equations but dotted lines are used for the relations with > or < signs and normal lines are used for those with ≥ or ≤ signs. Consider the inequalities x>0, y>0 and 2x + 3y >12 represented on the xy-plane In this case we draw the line x=0, y= 0 and 2x+3y=12 but the point about the inequality signs for each equation must be considered. From the figure above, the clear region satisfy all the inequalitiesx>0, y>0 and 2x + 3y >12, these three lines are the boundaries of the region. The Solution Set of Simultaneous Linear Inequalities Graphically Find the solution set of simultaneous linear inequalities graphically Example 5 Draw and show the half plane represented by 8x + 2y ≥16 Feasible Region Definition: In the xy plane the region that satisfies all the given inequalities is called the feasible region (F.R) Example 6 Indicate the feasible region for the inequalities 2x+3y ≥ 12 and y-x ≤ 2. Determine the solution set of the simultaneous inequalities y + x ≥3 and x-2y ≤ 9. Example 7 Fatuma was given 30 shillings to buy oranges and mangoes. An orange costs 2 shillings while a mango costs 3 shillings. If the number of oranges bought is at least twice the number of mangoes, show graphically the feasible region representing the number of ranges and mangoes she bought, assuming that no fraction of oranges and mangoes are sold at the market. Solution:- Let x be the number of oranges she bought and y the number of mangoes she bought. Now the cost of x and y together is 2x + 3y shillings which must not exceed 30 shillings. Inequalities: 2x + 3y ≤30 ……… (i) and x≥2y …………….. (ii), Also because there is no negative oranges or mangoes that can be bought, then x≥ and y≥0 ……….. (iii) Now the line 2x + 3y ≤30 is the line passing through (0, 10) and (15,0) and the line x≥2y or x – 2y ≥ 0 is the line which passes through (0,0) and (2,1). Exercise 2 For practice. Draw the graph of the equation 2x – y = 7 and show which half plane is represented by 2x – y >7 and the one represented by 2x – y 3 – x on the same axes and indicate the feasible region. A post office has to transport 870 parcels using a lorry, The Objective Function An Objective Function from Word Problems Form an objective function from word problems Linear programming components Any linear programming problem has the following: Objective Alternative course (s) of action which will achieve the objective. The available resources which are in limited supply. The objective and its limitations should be able to be expressed as either linear mathematical equations or linear inequalities. Therefore linear programming aims at finding the best use of the available resources. Programmingis the use of mathematical techniques in order to get the best possible solution to the problem Steps to be followed in solving linear programming problems; Read carefully the problem, if possible do it several times. Use the variables like x and y to represent the resources of interest. Summarize the problem by putting it in mathematical form using the variables let in step (b) above. In this step you need to formulate the objective function and inequalities or constraints. Plot the constraints on a graph From your graph, identify the corner points. Use the objective function to test each corner point to find out which one gives the optimum solution. Make conclusion after finding or identifying the optimum point among the corner points. Maximum and Minimum Values Corner Points on the Feasible Region Locate corner points on the feasible region Example 8 A student has 1200 shillings to spend on exercise books. At the school shop an exercise book costs 80shillings, and at a stationery store it costs 120 shillings. The school shop has only 6 exercise books left and the student wants to obtain the greatest number of exercise books possible using the money he has. How many exercise books will the student buy from each site? Therefore the student will buy 6 exercise books from each site. Example 9 A nutritionist prescribes a special diet for patients containing the following number of Units of vitamins A and B per kg, of two types of food f1 and f2 If the daily minimum in take required is 120 Units of A and 70 units of B, what is the least total mass of food a patient must have so as to have enough of these vitamins? Solution: Let x be the number of kg(s) of F1 that patient gets daily and y be the number of kg(s) of F2 to be taken by the patient daily. Objective function: F (x, y) = (x + y) minimum f (C) = 10 + 0 = 10 So f (B) = 6.8 is the minimum Therefore the least total mass of food the patient must have is 6.8 kilograms The Minimum and Maximum Values using the Objective Functio Find the minimum and maximum values using the objective function Example 10 A farmer wants to plant coffee and potatoes. Coffee needs 3 men per hectare while potatoes need also 3 men per hectare. He has 48 hired laborers available. To maintain a hectare of coffee he needs 250 shillings while a hectare of potatoes costs him 100 shillings. . Find the greatest possible land he can sow if he is prepared to use 25,000 shillings. Solution: Let x be the number of hectares of coffee to be planted and y be the number of hectares of potatoes to be planted. Objective function: f (x, y) = (x, + y) maximum 3x + 3y ≤ 48 or x + y ≤16 ………….(i) 250x + 100y≤ 25,000 Or 5x + 2y ≤ 500………(ii) x ≥ 0 ……………………(iii) y≥ 0 ……………………(iv) Using the objective function f (x, y) = (x + y) maximum, f (A) = (0 + 250) = 250 f (B) = (0+16) = 16 f (C) = (16+0) = 16 f(D) = (100+0)= 100 (maximum) Therefore the greatest possible area to be planted is 250 hectors of potatoes. NB: In most cases L.P problems must involve non-negativity constraints (inequalities) that are x ≥ 0 and y ≥ 0. This is due to the fact that in daily practice there is no use of negative quantities. Example 11 A technical school is planning to buy two types of machines. A lather machine needs 3m2 of floor space and a drill machine needs 2m2 of floor space. The total space available is 30m2. The cost of one lather machine is 25,000 shillings and that of drill machine is 30,000 shillings. The school can spend not more than 300,000 shillings, what is the greatest number of machines the school can buy? Solution: Let x be the number of Lather machines and y be the number of drill machines to be bought Objective function: f(x, y) = (x + y) max Inequalities: 3x + 2y ≤ 30.. ………………….(i) 25,000x + 30,000y ≤300,000 Or 5x + 6y ≤ 60……………………..(ii) x ≥ 0 ……………………………….(iii) y ≥ 0…………………… ………….(iv) Since the incomplete machine can’t work, then B = (8, 3) or (7, 4).That is approximating values of x and y to the possible integers without affecting the given inequalities or conditions. Now by using the objective function, f (A) = 0 + 10 = 10 f(B) = 7 + 4 0r f (B) = 8 + 3 = 11 f (C) = 10 + 0 = 10 f (D) = 0 + ) = 0 So f (B) gives the maximum number of machines which is 11. Therefore the greatest number of machines that can be bought by the school is 11 machines. Exercise 3 1. Show on a graph the feasible region for which the restrictions are: y ≤ 2x, x≥ 6, y≥2 and 2x + 3y ≤30 From the graph at which point does: y – x take a maximum value? x + y take a maximum value? y – x take a maximum value? 2. With only 20,000 shillings to spend on fish, John had the choice of buying two types of fish. The price of a single fish type 1 was 2,500shillings and each fish of type 2 was sold at 2,000 shillings. He wanted to buy at least four of type 1. What is the greatest number of fish did John buy? How many of each type could he buy? 3. How many corner points does the feasible region restricted by the inequalities? x≥0, y ≥ 0, 3x + 2y ≤ 18 and 2x + 4y ≤16 have? Which corner point maximizes the objective function f (x, y) = 2x + 5y?, Matrices and Transformation, Topic 6: Vectors - Basic Mathematics Form Four, Trigonometry, Probability, Three Dimensional Figures, Area and Perimeter, Coordinate Geometry, Statistics, Similarity
   

Topic 4: Probability – Basic Mathematics Form Four

Normally we are living in the world full of uncertainties.

For example when two equally strong foot ball teams play a match it is not easy to predicate the outcome of the game. Also for a pregnant woman it is not easy to predict what will be the sex of the born. Under such uncertainties the theory of probability is applied.

Definition; Probability is a branch of mathematics which deals with and shows how to measure the occurrence of events in daily life. Or it can simply be defined as a measure of chances.

Probability of an Events

The Probability of an Even Through Experiments

Determine the probability of an event through experiments

Probability set (s)

Definition: Probability set is the set of all outcomes/results from the experiment being performed.

For example when tossing once a fair coin the expected outcomes are either head(H) or tail(T) to be shown up.

In this case the probability set is

S = {H, T}

Also if a fair die is tossed once what is expected to show up is only one number among the six numbers, that is 1,2,3,4,5,6.

Now the probability set is

S = {1, 2, 3, 4, 5, 6}.

An event (E); An event is a specified outcome from the probability set.

For example a head (H) in the experiment of tossing a fair coin is an event and it is a sub set of the probability set,

Thus, S = {H, T} and E ={H}.

An event may or may not occur. For example if the event that a head occurs in tossing a fair coin once but a tail occurs instead , then the event did not occur and it is dented by E’ which is the complement of E.

So if S = {H, T} and the event E = {H}, then E’ is the event that H does not occur, hence E’ ={T}.

NB: A probability set is also called a sample space

Example 1

1. A dieis tossed once and the results are recorded. Find

  1. The probability set (sample space)
  2. The event that an even number occurs.
  3. The event that an even number does not occur.

Solution;

  1. The sample space S ={1,2,3,4,5,6}
  2. The event that an even number occurs is E = {2,4,6}.
  3. The event that an even number does not occur is E’ ={1,3,5}

Example 2

Give the probability set of the experiment of selecting even numbers less than 20.

Solution

S = {2, 4, 6, 8, 10, 12, 14, 16, 18}.

Example 3

Give the probability set of not selecting an even number from a set of counting numbers less than 9.

Solution

S= {1, 2, 3, 4, 5, 6, 7, 8}

E = {2, 4, 6, 8}

So E’ = {1, 3, 5, 7}

where E is the event of selecting an even number and E’ is the event of not selecting even number less than 9.

Exercise 1

1. Write theprobability set of each of the following experiments:

  1. A die is tossed and the face showing up is read.
  2. A friend is asked for the month of his birth.
  3. The sex of a human being is asked.
  4. A card is drawn from a box containing five cards bearing the numerals 2,4,6,8 and 10.

2. Write inset notation the elements of the following events:

  1. A fair die is rolled and the number obtained is greater or equal to 5.
  2. A prime number between 20 and 40 is chosen.

3. Write inset notation the elements of the event of not choosing an even number between 25 and 55

Experimental Results in Relation to Real Life Occurrences

Interpret experimental results in relation to real life occurrences

For example when tossing once a fair coin the expected outcomes are either head(H) or tail(T) to be shown up.

In this case the probability set is

S = {H, T}

Also if a fair die is tossed once what is expected to show up is only one number among the six numbers, that is 1,2,3,4,5,6.

Now the probability set is

S = {1, 2, 3, 4, 5, 6}.

The Formula for Finding the Probability of an Event

Write the formula for finding the probability of an event

Probability of an event:

Definition: The probability of an event is the ratio between the number of times the event has occurred to the total number of experiments that have been done.

If P(E) Is the probability of the event E, then

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Also the probability found by experimenting is referred to as experimental probability.

The Formula to Calculate the Probability of an Event

Apply the formula to calculate the probability of an event

Example 4

A drawing pin was tossed 1000 times. The number of tosses where the pin fell flat was 563. Calculate the probability that when such a pin is tossed, it will fall flat.

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Example 5

5% of torch bulbs manufactured by a certain factory were defective. What is the probability that when a bulb from that factory is tested it will be defective?

Solution:

P(E) = 5% = 5/100 = 0.05

Note that the probability of an event is defined under the condition that every outcome has an equal chance of occurring as other outcomes. Here we say the outcomes are equally likely or equiprobable.

Words like random selection, fair die and a fair coin are used mean that the choice is impartial (unbiased)

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Example 6

A piece of chalk is picked from a box containing 5 identical pieces two of which are red and the remaining are white. Find the probability that the piece of chalk picked is red

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Example 7

Find the probability that a ling appears in a drawing a single card from an ordinary deck of 52 cards.

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Example 8

What is the probability of not getting an even number when a fair die is tossed?

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Example 9

What is the probability of selecting a green ball from the box containing red and green balls if the probability of selecting red ball is 1/4?

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Example 10

When tossing a die what is the probability of getting a number greater or equal to 1?

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Exercise 2

For practice.

  1. Find the probability of choosing a number divisible by 2 from a set of numbers between 20 and 45.
  2. The total number of red and white pieces of chalk that are contained in a box is 20. How many pieces of white chalk are in the box if the probability of choosing a red piece of chalk is 2/5, given that the pieces are identical?
  3. What is the probability that a month selected at random from the twelve months of the year will have 31 days?
  4. A survey conducted at certain maternity ward showed that 60% of children born were female. What is the probability that Moses’ child, who was born in that ward is a male?
  5. A die was tossed 100 times, the six numbers with their frequency of occurrence were recorded in the following table:

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Combined Events

Experiments of Two Combined Events

Perform experiments of two combined events

If two or more simple events may occur or take place at the same time then the events are combined events.

For instance when the experiment of tossing two coins at the same time is done, then the event of interest can’t simply be determined.

Let E = {Obtaining two heads}

In this case there are two simple events which are obtaining the head on the first coin and obtaining the head on the second coin.

So E1 = {Obtaining the head on the first coin}

E2 = {Obtaining the head on the second coin}

Drawing a Tree Diagram of Combined Events

Draw a tree diagram of combined events

The event E can be found by using what is referred to as a tree diagram.

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Therefore, S = {(H, H), (H, T), (T, H), (T, T)} is the sample space.

Example 11

A die and Coin are tossed together. Draw a tree diagram to find the Sample space and hence determine the probability that a head and a number less than 3 occurs.

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Example 12

A fraction is written by selecting the numerator from the digits 1, 2, 3 and the denominator from the digits 6, 8.

  1. Draw a tree diagram to find the sample space of this experiment.
  2. Find the probability that a the fraction written is less than ½

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Example 13

In a family of 3 children what is the probability that

  1. All are girls
  2. At least two are boys

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Example 14

Three coins are tossed simultaneously. Find the probability that

  1. 3 heads appear
  2. 2 tails and one head appear

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Exercise 3

1. If two digitsnumeral is written choosing ten’s digits from the set {1, 2, 3, 4,} and the unit’s digit from {5,6} what is the probability that a number greater than 20 will appear?

2. A pair of dice istossed. Find the probability that the sum of the two numbers obtained is

  1. At least 8
  2. at most 1
  3. Exactly 6

3. In a familywith two children, what is the probability that:

  1. Both are boys
  2. At least one is a boy

4. A die and twocoins are tossed at the same time find the probability that

  1. A number 6 and two heads will appear.
  2. A number less than 4, a head and tail will appear.
  3. A number multiple of 2 and two tails will appear.

The Probability of Two Combined Events using the Formula

Find the probability of two combined events using the formula

Mutually Exclusive Events;

Two or more events are said to be mutually exclusive if the occurrence of one event hinders the occurrence of the other. This means that for mutually exclusive events, only one event may occur at a time, e.g., it is impossible for two numbers say 1 and 6 on a single die to show up for one tossing.

Therefore if A and B are two events, then the probability of A or B is given by

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Example 15

If in a class there are 34 students instead of 35 and Issa, anna, Eliza and Juma apply for the one chance remaining what is the probability that either Anna or Juma will be chosen?

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Example 16

Find the probability that an even or an odd number which is greater than 1 occurs when a die is tossed once.

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Example 17

The following table shows years of experience for plumbers in a builders company.

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Independent Events

Definition: Two or events are said to be independent events if the occurance of one event does not affect the occurrence of other event(s)

For example
when a die and a coin are tossed together, the occurrence of a tail on the coin does not hinder the occurrence of the number 5 on the die.

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Example 18

A die and coin are tossed. Find the probability that a number greater than 4 appears on the die and a tail appears on the coin:

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Example 19

A box contains 9 oranges, 7 mangoes and 2 lemons. A fruit is drawn from the box and then replaced. Another draw is made. What is the probability that both fruits drawn are mangoes.

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Example 20

The probability that a man and his wife will be alive for 50 years are 3/10 and 1/3 respectively.
Find the probability that

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Exercise 4

Self test.

  1. A coin is tossed and a card is drawn from an ordinary pack of 52cards.Find the probability that an ace is drawn and a head is obtained on the coin (There 4 aces in a pack of cards)
  2. Two numbers are selected from the integers 1 to 11 inclusively, repeation being allowed. Find the probability that (a) Both prime (b) Both are powers of 2

3. In the village,the probability that a man selected at random on a Sunday morning is carrying more than is 0.7. Find the probability that;

  1. Two men selected at random on a Sunday morning is carrying more than 30kg
  2. Three men selected at random are all carrying more than 30kg

4. A letter is chosenfrom the word “random” What is the probability that it is an n or d?

5. (a) What does itmean by saying that the probability of an event is (i) 0 (ii) 1 (b) Give two examples of impossible of events.

The Knowledge of Probability to Determine the Occurrence of Events in Real Life Situation

Apply the knowledge of probability to determine the occurrence of events in real life situation

Probability is an area of mathematics which we use all the time in daily life – and usually without thinking about it. While many aspects are very intuitive, probabilities may be different for different people. I might estimate that the chance of rain is 70%, while a meteorologist with detailed weather data might say the chance of rain is 64.2%.

Activity 1: Probability 

Apply the knowledge of probability to determine the occurrence of events in real life situation.

Related Posts:

  1. Topic 1: Coordinate Geometry – Basic Mathematics Form Four
  2. Topic 2: Area and Perimeter – Basic Mathematics Form Four
  3. Topic 5: Trigonometry – Basic Mathematics Form Four

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