Learning Outcomes: Discuss compound events & mutually exclusive events define the concept of experiments, outcomes, sample space, events and random selections. (a) (b) state the basic laws of probability. (c) find the probability of an event. (d) find the probabilities of the intersections and union of two events. Use of Venn diagrams, tree diagrams & contingency table to solve problems (e) Find conditional probability and independent events. At the end of the lesson, students should be able to a process that, when it is performed, it results in one and only one of many possible results that cannot be predicted with certainty. Experiment the possible results obtained from an experiment. Basic Outcomes the set of all the possible outcomes obtained from an experiment Sample space, S [denoted by capital letter “S”] an event is a subset of basic outcomes from the sample space Events [Usually denoted by a capital letter ] Example 1 Experiment : Tossing a coin Basic Outcomes: Head (H) , Tail (T) Sample space: Example 2 Experiment : Rolling a die Basic Outcomes: Sample space: Event of getting odd numbers: 7.1 Probability C07 - SM025 Page 1
The sample space when two dice are tossed or one die is tossed twice is given as follows : The number of outcomes in sample space, Let be the event of getting the sum of 2 numbers is 6 The number of outcomes in event , Let be the event of getting the sum of two numbers is a multiple of 5 The number of outcomes in event , Example 3 C07 - SM025 Page 2
are mutually exclusive Event and event are said to be mutually exclusive events if there is NO intersection between the 2 events. Example 4 Rolling a die : getting odd numbers Let : getting even numbers Since , are mutually exclusive events. Remarks: For 3 events [events ] are said to be mutually exclusive events, all the events will have NO intersection at all as shown in diagram below: Mutually Exclusive Events C07 - SM025 Page 3
are exhaustive events Event and event are exhaustive events if the union of those events forms the sample space. Remarks: Exhaustive events may have intersection, i.e. they may be NOT mutually exclusive. Example 5 (i) Given Since , and are exhaustive events. (ii) Given Since , and are NOT exhaustive events. Exhaustive Events C07 - SM025 Page 4
The complementary of event [denoted by ] refers to the event of all outcomes in sample space that are NOT in event . • sometimes interpreted as “Event A DOES NOT occur” • [Complementary Events are Exhaustive Events] • [Complementary Events are Mutually Exclusive Events] Remarks: Example Given (a) (b) prime numbers NOT prime number (c) odd numbers even number Complementary Events C07 - SM025 Page 5
The probability of an event is the possibility (in numerical) of that event to occur. , The probability of the event [denoted by ] is given by where the number of outcomes in event the number of outcomes in sample space, • Event will surely (100%) to occur • Event is impossible (0%) to occur • Remarks: Example 6 (a) the sum of the two numbers obtained is 8. (b) the sum of the two numbers obtained is a prime number. When two dice are tossed, find the probability Solutions: (a) Let the event of the sum of the two numbers obtained is 8. (b) Let the event of the sum of the two numbers obtained is a prime number. The Probability of an Event C07 - SM025 Page 6
(a) Illustrate the above information in a Venn diagram. (i) both Biology and Chemistry (ii) either Biology or Chemistry or both If one of those student is randomly chosen, find the probability that the student is studying (b) There are 100 matriculation students, of whom 20 are studying Biology, 15 are studying Chemistry and 8 are studying both Biology and Chemistry. Solutions: (a) Let study Biology study Chemistry (b)(i) (b)(ii) Example 7 C07 - SM025 Page 7
The probability of the event or event occuring (also both events occurring), i.e. is given by Example 1 In a class of 30 students, 5 out of the 20 girls and 7 out of the 10 boys are members of the Mathematics Society. A student is selected randomly. What is the probability that the student is a girl or a member of the Mathematics Society. Solution: Let the event of the student is a girl the event of the student is a member of the Math Society Probability of 2 Events C07 - SM025 Page 8
An integer is selected randomly from a set of integers as follows: . Find the probability that the integer is (a) an even number or is divisible by 3 (b) an even number and is not divisible by 3 (c) not an even number and is not divisible by 3. Solutions: Let the event of the integer is an even number the event of the integer is divisible by 3 (a) (b) (c) Example 2 C07 - SM025 Page 9
Information usually given in "Quantity" The events can be put into 2 categories (row & column) Events in each category must be mutually exclusive. A contingency table is a tabular representation of data that can facilitate calculating probabilities. Events Total Total Events in Rows: Events are mutually exclusive Events in Columns: Events are mutually exclusive Contingency table enable probabilities of some events can be obtained by simple calculation E.g. Find using instead of Information in table can be also given in "Probability" Events Total Total Remarks: Contingency Table C07 - SM025 Page 10
A survey is conducted on a group of workers comprising of production operators, administrative officers and security guards. The survey is to determine the total working hours in a week. Working hour per week Production Operator (R) Administrative Officer (A) Security guard (G) Total hrs (L) 63 21 4 88 hrs (M) 47 14 10 70 hrs (H) 87 8 17 112 Total 196 43 31 270 One of the workers in the survey is randomly selected. (a) being a production operator. Based on the information provided, calculate the probability of the worker Solution: (b) work between 40 – 70 hrs. Solution: (c) being an administrative officer and working more than 70 hours. Solution: (d) being a security guard working less than 40 hours. Solution: Example 3 C07 - SM025 Page 11
A Venn diagram is an illustration that uses circles and other shapes to show the logical relation between sets. Very useful to derive formulae for probability calculation [In probability, it is used to show that relation between events.] Venn Diagram C07 - SM025 Page 12
Probability involving Mutually Exclusive Events Given events and are mutually exclusive events, then both the events are impossible to occur at the same time and therefore the probability of both events occur at the same time is zero. which will further implies that Remarks: For 3 events [events ] being mutually exclusive events, Probability involving Complementary Events Given events and are complementary events, then event is the event where event does not occurs. which also implies that • • • Remarks: Probability involving Mutually Exclusive Events & Complementary Events C07 - SM025 Page 13
Given and are two events where , and . (a) Determine whether and are two mutually exclusive events. Solution: Since and are two mutually exclusive events. Find and (b) Solution: Example 4 C07 - SM025 Page 14
Conditional probability refers to the probability of an event to occur with the condition another event had already occurred. Consider events in a sample sample, . The probability of event occurs with the condition event had already occurred [denoted as ] is given by [ is read as the probability of given ] The condition event [event B] usually refer to the event come after the words "if ", "given that" , "knowing that" and etc If event B occurs, then find the probability of event occurs. Find the probability of event occurs if event B occur Find the probability of event occurs given that event B occurs. Given that event B occurs, find the probability of event occurs. Find the probability of event occurs knowing that event B occurs. It is known that event B occurs, find the probability of event occurs. Common statements used in question to find • can also be given in "quantity": may also interpret as the probability of the occurrence of event within event instead of the sample space. • Remarks: Consider Sample space, = KMPP students Event Lecture group H3 students Event Female students Conditional Probability C07 - SM025 Page 15
and are two events such that , and . Find Solutions: Solutions: Solutions: Example 1 C07 - SM025 Page 16
30 professors out of 100 who are examined were found to be overweight (W). 10 of them had high blood pressure (H). Only 4 of the professors who were not overweight had high blood pressure. Find the probability that a professor (a) is overweight if he had high blood pressure, (b) will not have high blood pressure if he is overweight. Events Total Total Solutions: (a) (b) Example 2 C07 - SM025 Page 17
Usually apply for conditional probability & events with sequels when solving probability question. • E.g. Consider events in a sample sample, . Tree Diagram is diagram that represent a sequence of events. For a tree diagram with complete information, the sum of probabilities of all events in each of branch must be equal to 1. • Remarks: more than 2 events in a branch more than 2 branches ONLY one event in a branch (starting from the 2nd branch) In some of the questions, there might be Tree Diagram C07 - SM025 Page 18
Harry travels to work by either route A or route B. (a) What is the probability that he is late for work on a particular day? Given that he is not late for work, what is the probability that he chooses route B ? (b) The probability that he chooses route A is . The probability that he is late for work if he chooses route A is and the probability that he is late for work if he chooses route B is . Solutions: Let choose Route A choose Route B late for work (a) (b) Example 3 C07 - SM025 Page 19
are independent events Event and event are said to be independent events if the outcomes of the two events does not affect each other. are independent events which also implies Example 4 Suppose two events A and B are independent. Given P(A) = 0.4 and P(B) = 0.25. Find (a) A and B are independent. Solution: (b) Solution: Independent Events C07 - SM025 Page 20
, and are three events such that and are independent whereas and are mutually exclusive. Given , , and . Solution: A and B are independent. Solution: Solution: Example 5 C07 - SM025 Page 21
A puzzle is given to two students Aziz and Bong. From past experience, it is known that the probabilities Aziz and Bong will get the correct solution are 0.65 and 0.6 respectively. If they attempt to solve the puzzle without consulting each other, (a) the puzzle will be solved correctly by both of them. (b) only one of them will get the correct solution. find the probability that Let Aziz get correct solution Bong get correct solution Solutions: (a) (b) Example 6 C07 - SM025 Page 22