UNIT 5: CONDITIONAL PROBABILITY AND BAYES THEOREM
Key unit Competence: Apply rules of probability to solve problems relatedto dependent and independent events.
5.0 INTRODUCTORY ACTIVITY
1) Consider a machine which manufactures electronic components. These
must meet certain specification. The quality control departmentregularly
samples the components.
Suppose, on average, 92 out of 100 components meet the specification.
Imaginethata
Componentisselected at random and let A be the outcome that a component
meets the specification; let B be the outcome that a component does not
meet the specification.
a) Explain in your own words and determine the probability that a
components meet the specification.
b) Explain in your own words and determine the probability that a
component does not meet the specification.
2) A box contains 4 whitechalks and 3 black chalks. One chalk is drawn at
random; its color is noted but not replaced in the box.
a) What is the probability of selecting2 white chalks?b) Determine the probability of selecting 3 white and 2 black chalks.
Probability is a measure of the likelihood of the occurrence of a particularoutcome.
5.1Tree diagram
ACTIVITY 5.1
A box contains 4 blue pens and 6 black pens. One pen is drawn at
random, its color is noted and the pen is replaced in the box. A pen is
again drawn from the box and its color is noted.
1) For the 1st trial, what is the probability of choosing a blue pen
and probability of choosing a black pen?
2) For the 2nd trial, what is the probability of choosing a blue pen
and probability of choosing a black pen? Remember that after
the 1st trial the pen is replaced in the box.
3) In the following figure complete the missing colors andprobabilities
CONTENT SUMMARY
A tree diagram is one way of illustrating the probabilities of certain outcomes
occurring when two or more trials take place in succession by use of arrows in
the form of a tree and branches. A tree diagram has branches and sub-branches
which help us to see the sequence of events and all the possible outcomes ateach stage.
The outcome is written at the end of the branch and the fraction on the branch
gives the probability of the outcome occurring.
For each trial the number of branches is equal to the number of possible
outcomes of that trial.
Examples:
1) A bag contains 8 balls of which 3 are red and 5 are green. One ball is drawn at
random, its colour is noted and the ball replaced in the bag. A ball is again drawn
from the bag and its colour is noted. Find the probability that the 2 balls drawn
will be
a) red followed by green,
b) red and green in any order,c) of the same colour.
APPLICATION ACTIVITY 5.1
1. Calculate the probability of three coins landing on: Three heads.
2. A class consists of six girls and 10 boys. If a committee of three is
chosen at random, find the probability of:
a) Three boys being chosen.
b) Exactly two boys and a girl being chosen.
c) Exactly two girls and a boy being chosen.
d) Three girls being chosen.
3. A bag contains 7 discs, 2 of which are red and 5 are green. Two discs
are removed at random and their colors noted. The first disk is not
replaced before the second is selected. Find the probability that
the discs will be:
a) both red b) of different colors c) the same colors.
4. Three discs are chosen at random, and without replacement, from a
bag containing 3 red, 8 blue and 7 white discs. Find the probability
that the discs chosen will bea) all red b) all blue c) one of each color.
5.2The Addition law of probability
ACTIVITY 5.2
Consider a machine which manufactures car components. Suppose
each component falls into one of four categories:top quality,
standard, substandard, reject
After many samples have been taken and tested, it is found that
under certain specific conditions the probability that a componentfalls into a category is as shown in the following table.
5.3 Independent events
ACTIVITY 5.3
A box contains 3 red pens, 4 green pens and 5 blue pens. One pen is
taken from the box and then replaced. Another pen is taken from the
box. Let A be the event “the first pen is red” and B be the event the
second pen is blue.”
Is the occurrence of event B affected by the occurrence of event A?Explain.
5.4.Dependent events
ACTIVITY 5.4
Suppose that you have a deck of 52 cards. You can draw a card from
that deck , continue without replacing it, and then draw a second card .
a) What is the sample space for each event?
b) Suppose you select successively two cards, what is the probability
of selecting two red cards?
c) Explain if there is any relationship (Independence or dependence)
between those two events considering the sample space. Does the
selection of the first card affect the selection of the second card?
When the outcome or occurrence of the first event affects the outcome or
occurrence of the second event in such a way that the probability is changed,
the events are said to be dependent.
Examples:
1)Suppose a card is drawn from a deck and not replaced, and then the second
card is drawn. What is the probability of selecting an ace on the first card and aking on the second card?
Note that:
The event of getting a king on the second draw given that an ace was drawn the
first time is called a conditional probability.
APPLICATION ACTIVITY 5.4
The world wide Insurance Company found that 53% of the residents
of a city had home owner’s Insurance with its company of the clients,
27% also had automobile Insurance with the company. If a resident
is selected at random, find the probability that the resident has
both home owner’s and automobile Insurance with the world wideInsurance Company.
5.5 Conditional probability
ACTIVITY 5.5
A box contains 3 red pens, 4 green pens and 5 blue pens. One pen is
taken from the box and is not replaced. Another pen is taken from
the box. Let A be the event “the first pen is red” and B be the event
“the second pen is blue.”
Is the occurrence of event B affected by the occurrence of event A?Explain.
Examples:
1) A die is tossed. Find the probability that the number obtained is a 4 giventhat the number is greater than 2.
2) At a middle school, 18% of all students play football and basketball, and 32%
of all students play football. What is the probability that a student who playsfootball also plays basketball?
Notice:
Contingency table
Contingency table (or Two-Way table) provides a different way of calculating
probabilities. The table helps in determining conditional probabilities quite
easily. The table displays sample values in relation to two different variables
that may be dependent or contingent on one another.
Below, the contingency table shows the favorite leisure activities for 50 adults,
20 men and 30 women. Because entries in the table are frequency counts, thetable is a frequency table.
Calculate the following probabilities using the table:
a) P(person is a car phone user)
b) P(person had no violation in the last year)
c) P(person had no violation in the last year AND was a car phone user)
d) P(person is a car phone user OR person had no violation in the last year)
e) P(person is a car phone user GIVEN person had a violation in the last year)
f) P(person had no violation last year GIVEN person was not a car phone user)
b. The respondent was a male, given that the respondent answered no.
APPLICATION ACTIVITY 5.5
The world wide Insurance Company found that 53% of the residents
of a city had home owner’s Insurance with its company of the clients,
27% also had automobile Insurance with the company. If a resident
is selected at random, find the probability that the resident has
both home owner’s and automobile Insurance with the world wide
Insurance Company.
1. A jar contains black and white marbles. Two marbles are chosen without
replacement. The probability of selecting a black marble and then a white
marble is 0.34, and the probability of selecting a black marble on the first
draw is 0.47. What is the probability of selecting a white marble on the
second draw, given that the first marble drawn was black?
2. A bag contains five discs, three of which are red. A box is contains six discs,
four of which are red. A card is selected at random from a normal pack of
52 cards, if the card is a club a disc is removed from the bag and if the card
is not a club a disc is removed from the box. Find the probability that, if the
removed disc is red it came from the bag.
3. The probability that Gerald parks in a no-parking zone and gets a parking
ticket is 0.06, and the probability that Gerald cannot find a legal parking
space and has to park in the no-parking zone is 0.20. On Tuesday, Gerald
arrives at Headquarter office and has to park in a no-parking zone. Find theprobability that he will get a parking ticket.
APPLICATION ACTIVITY 5.6
#
1. 20% of a company’s employees are engineers and 20% are
economists. 75% of the engineers and 50% of the economists hold
a managerial position, while only 20% of non-engineers and noneconomists
have a similar position. What is the probability that
an employee selected at random will be both an engineer and a
manager?
2. The probability of having an accident in a factory that triggers
an alarm is 0.1. The probability of its sounding after the event
of an incident is 0.97 and the probability of it sounding after no
incident has occurred is 0.02. In an event where the alarm has
been triggered, what is the probability that there has been noaccident?
5.7 END UNIT ASSESSMENT
1) The probability that it is Friday and that a student is absent is
0.03. Since there are 5 school days in a week, the probability that
it is Friday is 0.2. What is the probability that a student is absent
given that today is Friday?
2) Dr. Richard is conducting a survey of families with 3 children.
If a family is selected at random, what is the probability that
the family will have exactly 2 boys if the second child is a boy?
Assume that the probability of giving birth to a boy is equal to the
probability of giving birth to a girl.
3) A 12-sided die (dodecahedron) has the numerals 1 through 12 on
its faces. The die is rolled once, and the number on the top face is
recorded. What is the probability that the number is a multiple of
4 if it is known that it is even?
4) At Kennedy Middle School, the probability that a student takes
Technology and Spanish is 0.087. The probability that a student
takes Technology is 0.68. What is the probability that a student
takes Spanish given that the student is taking Technology?
5) A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn
at random. What is the probability that none of the balls drawnis blue?
6) In a certain college, 5% of the men and 1% of the women are taller
than 180 cm. Also, 60% of the students are women. If a student is
selected at random and found to be taller than 180 cm, what is theprobability that this student is a woman?
REFERENCES
1. J. Sadler, D. W. S. Thorning: Understanding Pure Mathematics, Oxford
University Press 1987.
2. Arthur Adam Freddy Goossens: Francis Lousberg. Mathématisons 65.
DeBoeck,3e edition 1991.
3. Charles D. Hodgman, M.S., Samuel M. Selby, Robert C.Weast. Standard
Mathematical Table. Chemical Rubber Publishing Company, Cleveland, Ohio
1959.
4. David Rayner, Higher GCSE Mathematics, Oxford University Press 2000
5. Direction des Progammes de l’Enseignement Secondaire. Géometrie de
l’Espace 1er Fascule. Kigali, October1988
6. Direction des Progammes de l’Enseignement Secondaire. Géometrie de
l’Espace 2ème Fascule. Kigali, October1988
7. Frank Ebos, Dennis Hamaguchi, Barbana Morrison & John Klassen,
Mathematics Principles & Process, Nelson Canada A Division of International
Thomson Limited 1990
8. George B. Thomas, Maurice D. Weir & Joel R. Hass, Thomas’ Calculus Twelfth
Edition, Pearson Education, Inc. 2010
9. Geoff Mannall & Michael Kenwood, Pure Mathematics 2, Heinemann
Educational Publishers 1995
10. H.K. DASS...Engineering Mathematics. New Delhi, S. CHAND&COMPANY
LTD, thirteenth revised edition 2007.
11. Hubert Carnec, Genevieve Haye, Monique Nouet, ReneSeroux, Jacqueline
Venard. Mathématiques TS Enseignement obligatoire. Bordas Paris 1994.
12. James T. McClave, P.George Benson. Statistics for Business and Economics.
USA, Dellen Publishing Company, a division of Macmillan, Inc 1988.
13. J CRAWSHAW, J CHAMBERS: A concise course in A-Level statistics with
worked examples, Stanley Thornes (Publishers) LTD, 1984.
14. Jean Paul Beltramonde, VincentBrun, ClaudeFelloneau, LydiaMisset, Claude
Talamoni. Declic 1re S Mathématiques. Hachette-education, Paris 2005.
15. JF Talber & HH Heing, Additional Mathematics 6th Edition Pure & Applied,
Pearson Education South Asia Pte Ltd 1995
16. J.K. Backhouse, SPTHouldsworth B.E.D. Copper and P.J.F. Horril. Pure
Mathematics 2. Longman, third edition 1985, fifteenth impression 1998.
17. Mukasonga Solange. Mathématiques 12, AnalyseNumérique. KIE, Kigali
2006.
18. N. PISKOUNOV, Calcul Différential et Integral tom II 9ème édition. Editions
MIR. Moscou, 1980.
19. Paule Faure- Benjamin Bouchon, Mathématiques Terminales F. Editions
Nathan, Paris 1992.
20. Peter Smythe: Mathematics HL & SL with HL options, Revised Edition,
Mathematics Publishing Pty. Limited, 2005.
21. Robert A. Adms & Christopher Essex, Calculus A complete course Seventh
Edition, Pearson Canada Inc., Toronto, Ontario 2010
22. Seymour Lipschutz. Schaum’s outline of Theory and Problems of Finite
Mathematics. New York, Schaum Publisher, 1966
23. Seymour Lipschutz. Schaum’s outline of Theory and Problems of linear
algebra. McGraw-Hill 1968.
24. Shampiyona Aimable : Mathématiques 6. Kigali, Juin 2005.
25. Yves Noirot, Jean–Paul Parisot, Nathalie Brouillet. Cours de Physique
Mathématiques pour la Physique. Paris, DUNOD, 1997.
26. Swokowski, E.W. (1994). Pre-calculus: Functions and graphs, Seventh
edition. PWS Publishing Company, USA.
27. Allan G. B. (2007). Elementary statistics: a step by step approach, seventh
edition, Von Hoffmann Press, New York.
28. David R. (2000). Higher GCSE Mathematics, revision and Practice. Oxford
University Press, UK.
29. Ngezahayo E.(2016). Subsidiary Mathematics for Rwanda secondary
Schools, Learners’ book 4, Fountain publishers, Kigali.
30. REB. (2015). Subsidiary Mathematics Syllabus, MINEDUC, Kigali, Rwanda.
31. REB. (2019). Mathematics Syllabus for TTC-Option of LE, MINEDUC, Kigali
Rwanda.
32. Peter S. (2005). Mathematics HL&SL with HL options, Revised edition.
Mathematics Publishing PTY. Limited.
33. Elliot M. (1998). Schaum’s outline series of Calculus. MCGraw-Hill
Companies, Inc. USA.
34. Frank E. et All. (1990). Mathematics. Nelson Canada, Scarborough, Ontario
(Canada)
35. Gilbert J.C. et all. (2006). Glencoe Advanced mathematical concepts,
MCGraw-Hill Companies, Inc. USA.
36. Robert A. A. (2006). Calculus, a complete course, sixth edition. Pearson
Education Canada, Toronto, Ontario (Canada).
37. Sadler A. J & Thorning D.W. (1997). Understanding Pure mathematics,
Oxford university press, UK.
38. J. CRAWSHAW and J. CHAMBERS 2001. A concise course in Advanced Level
Statistics with worked examples 4th Edition. Nelson Thornes Ltd, UK.
39. Ron Larson and David C (2009). Falvo. Brief Calculus, An applied approach.
Houghton Mifflin Company.
40. Michael Sullivan, 2012. Algebra and Trigonometry 9th Edition. Pearson Education, Inc
41. Swokowski & Cole. (1992). Preaclaculus, Functions and Graphs. Seventh edition.
42. Glencoe. (2006). Advanced mathematical concepts, Precalculus with Applications.
43. Seymour Lipschutz, PhD. & Marc Lipson, PhD. (2007). Discrete mathematics.3rd edition.
44. K.A. Stroud. (2001). Engineering mathematics. 5th Edition. Industrial Press,Inc, New York
45. John bird. (2005). Basic engineering mathematics. 4th Edition. LinacreHouse, Jordan Hill, Oxford OX2 8DP