baiermathstudies.wikispaces.com · Web view2012-03-08 · 1. Topics 3.1 & 3.3 . Set Theory. ... Often probability questions can be answered by drawing a Venn diagram to represent

Topic 3 Revision Notes Sets, Logic & Probability IB Math Studies

Topics 3.1 & 3.3 Set Theory

Set NotationSymbol Definition Means…

U universal set All the elements given in a question is an element of Is an element or member of the set is not an element of Is not an element or member of the set

‘ Complement opposite of the set the empty set an empty set with no members/elements Intersection An overlap; only the elements in both sets.

Union A marriage; all the elements in each set, including all elements that are in both.

Subset A smaller set contained wholly within a larger set

n(A) Number of elements inside set A

Topic 3.2 Venn diagrams and sets

Often probability questions can be answered by drawing a Venn diagram to represent the problem. Below are some sample Venn diagrams

Union Intersection

Subsets Mutually Exclusive Events

(A B)’ A’ B

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(A B)’ A’ B

Guided exampleA group of 40 IB students were surveyed about the languages they have chosen at IB: E = English, F = French, S = Spanish.

3 students did not study any of the languages above.2 students study all three languages8 study English and French10 study English and Spanish6 study French and Spanish13 students study French28 students study English

(a) Draw a Venn diagram to illustrate the data above. On your diagram write the number in each set.(b) How many students study only Spanish?(c) On your diagram shade (E F)’ , the students who do not study English or French.

Answer (a)The key point here is that when it says, “8 study English and French” it did NOT say, “8 study English and French only.” So 2 the students that study all three languages have been counted again in the 8 who study English and French. Hence, only 6 students study English and French only.

• The final diagram will look like this:

Answer (b)By looking at the Venn diagram we can clearly see that we have 4 who do Spanish only.

Answer (c)By shading E F we would have:Therefore (E F)’ will be the opposite of this shading.

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Topics 3.4 through 3.7 Logic and truth tables

Propositions – A statement that can be either true or false. For example, the baby is a girl.

Negation – A proposition that has become negative: For example, the baby is not Australian. Be careful that it is a negative and not the opposite. The symbol used is ¬.

p ¬pT FF T

Conjunction – the word ‘and’ used to join two conjunctions together. For example, the baby is a girl and Australian. The symbol used is .

p q p qT T TT F FF T FF F F

Disjunction – the word ‘or’ used to join the conjunctions together. For example, the baby is a girl or Australian. The symbol used is .

p q p qT T TT F TF T TF F F

Exclusive Disjunction - is true when only one of the propositions is true. The word ‘or’ is used again, but in this case you must write “or, but not both” For example,p: Sally ate cereal for breakfastq: Sally ate toast for breakfast

p q p qT T FT F TF T TF F F

Implication – Using the words if …… then with two propositions. For example: If you do not sleep tonight then you will be tired tomorrow. The symbol used for implication is .

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p q p qT T TT F FF T TF F T

By using the implication we can generate the converse, inverse, and contrapositive.

For example,p: My shoes are too small. q: My feet hurt.

Implication: p q. If my shoes are too small, then my feet hurt. Converse: q p. If my feet hurt, then my shoes are too small. Inverse: p q. If my shoes are not too small, then my feet do not hurt. Contrapositive: q p. If my feet do not hurt, then my shoes are not too small.

Equivalence – If two propositions are linked with “… if and only if…”, then it is an equivalence. p q is the same as p q and q p. Or using notation: (p q) = (p q) (q p)

p q p qT T TT F FF T FF F T

Tautology – a statement that produces True (T) throughout the column of the truth table.

Contradiction – a statement that produces False (F) throughout the column of a truth table.

Guided exampleLet the propositions p, q, and r be defined as:

p: Andrea studies IB English.q: Andrea studies IB Spanish.r: The school offers at least 2 IB languages.

(a) Write the following in logical form. If Andrea studies English and Spanish, then the school offers at least 2 languages.(b) Write the following statement in words: ¬p ¬q(c) Copy and complete the truth table below.

P Q r p q (p q) (p q) rT T TT T FT F TT F FF T TF T FF F TF F F

(d) Is (p q) r a tautology, contradiction or neither?

Answer (a)

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The words if … then are in the statement so it must be: (p q) r Answer (b)

If Andrea does not study English then she will not study Spanish.Answer (c)

Although these truth tables can appear daunting at first, taking each column one at a time makes things quite simple.

The final truth table will be:p q r p q (p q) (p q) rT T T T F TT T F T F TT F T F T TT F F F T FF T T F T TF T F F T FF F T F T TF F F F T F

Answer (d) (p q) r is neither a tautology nor a contradiction because the final column is neither all T’s

or all F’s.

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Topics 3.8 to 3.10 ProbabilitySuppose a fair die is rolled. Then there are six possible outcomes, 1 2 3 4 5 6

Suppose you want to know the likelihood of getting a 3. There is only one way of getting a 3 so .The list of all possible outcomes is called the sample space, and those outcomes which meet the particular requirement are called the event.

The following formulae are given in the IB formula booklet

Probability of an event A: P( A )=n( A )n(U )

Complementary events: P(A’) = 1 – P(A)

Combined events: P( A∪B )=P( A )+P(B )−P( A∩B)

Mutually exclusive events: P( A∪B )=P( A )+P(B )

Independent events

Conditional probability

where n(A) is the number of elements in the set A, and n(U) is the number of elements in the whole sample space U.Probability of an event

The probability of event must be written as a number between 0 and 1, and written as a fraction, percentage or decimal. It is easiest to use a fraction, if possible.

The first formula means put the number of times A occurs as the numerator and the number of all the events as the denominator.

Complementary events

If the probability of A is

13 then the probability of A’ (not A) is 1 –

13 =

23 .

Combined events Combined events are events that CAN happen at the same time (for instance choosing a face card or a

heart from a standard deck of playing cards). This formula can be understood by using a Venn diagram. Add up the number of times A occurs with

the number of times B occurs and subtract the number of times both occur together. For EXAMPLE: in a class of 140 students if 80 people chose Biology, 60 chose Chemistry and 20 chose

both Biology and Chemistry, then P(AB) =

80140

+60140

−20140

=120140 .

This can be shown with a Venn diagram:

20

Bio Ch

60 4020

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GUIDED EXAMPLE

Given P(A) = 0.55, P( A∪B )=0 .7 and P( A∩B )=0 . 2 , find P(B’).SOLUTION:P( A∪B )=P( A )+P(B )−P( A∩B)

0.7 = 0.55 + P(B) – 0.20.35 = P(B)

∴P(B’)=1 – P(B) P(B’) = 1 – 0.35 = 0.65

Mutually Exclusive events Mutually exclusive events are events that CANNOT happen at the same time (for instance choosing a 7

or a face card from a standard deck of playing cards). You can think of mutually exclusive events as being a special case of combined events where

P(A∩B)=0. So the combined events formula P(AB)=P(A)+P(B) – P(AB) becomes P(AB)=P(A)+P(B).

Much of probability is based on set notation. Looking at the Venn diagram below we can see the numbers that represent the various parts of the formulae

using this information we can give the following probabilities:

P(A) =

3050 P(A’) =

1−3050

=2050 P(AB)=

3050

+3550

−2050

=4550 P(A│B) =

2035 P(B│A) =

2030

Independent events

The term “independent” is equivalent to “statistically independent.” Statistically independent is similar, although not exactly, as the everyday usage of the word ‘independent’ in the sense that independent events are events that have nothing to do with each other. For EXAMPLE, the event of flipping a coin and tossing a die.

P(A∩B)=P(A)P(B) for independent events. By definition the following is true for independent events: P(A|B)=P(A)=P(A|B′).

Conditional probability

This is used when we see the ‘given that’ in the question. P(A/B) means the probability of A given B. When we are ‘given’ something in probability questions we can cut down the outcomes we have to only

things where B has occurred. This forms the denominator of the fraction. The numerator of the fraction is where A and B have occurred.

5

A B

10 1520

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An easy way to answer the “given that” questions is with a Venn diagram, if possible. Whatever follows the “given that” becomes the sample space or the bottom of the fraction.

GUIDED EXAMPLEAn unbiased 6-sided die is marked with the numbers: 2, 2, 3, 4, 5, and 8.The die is rolled. Find the probability of the die:

(a) landing on a prime(b) not landing on a 4(c) landing on a 2 given that it has landed on a prime.

ANSWER

Probability from tables

Frequently probability questions are often given in tables. These questions are reasonably easy to answer. The conditional probability question is the only one that sometimes confuses students.

EXAMPLEThe table below shoes the choices made by males and females for their IB second language.

Female Male TotalFrench 60 20 80Spanish 50 70 120

Total 110 90 200A student is chosen at random. Find the probability that the student is:(a) female,(b) male and studies French,(c) a female, given that the student studies Spanish.

ANSWER(a) The total is 200 so that will become the denominator of the fraction. There are 110 females so the

answer is:

110200 .

(b) There are 20 students who are male and study French so the answer is

20200

(c) The ‘given that’ makes this question a conditional probability question. For the denominator we just need the total number of students who study Spanish. The numerator would be the number of

females who study Spanish. So the answer is:

50120 = 0.417.

The problem above can also be answered by using the formula P( A|B )= P( A∩B )

P (B ) ⇒

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, 04/04/11,


P( female|Spanish)=P( female and studies Spanish )

P (Spanish) ⇒

P( female|Spanish )=

50200120200

Tree diagramsTree diagrams are a clear way to view all the outcomes and their associated probabilities when 2 or more events occur. From a tree diagram you can find simple probabilities or even a conditional probability.

EXAMPLEWhen Geraldine travels to work she can travel either by car (C), bus (B) or train (T). She travels by car on one day in five. She uses the bus 50 of the time. The probabilities of her being late (L) when travelling by car, bus or train are 0.05, 0.12 and 0.08 respectively.

Complete the tree diagram below, where NL represents not late.

L

L

LN L

N L

N L

C

B

T

0 .2

a) Find the probability that Geraldine travels by bus and is late.

b) Find the probability that Geraldine is late.

c) Find the probability that Geraldine travelled by train, given that she is late.

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, 04/04/11,

Math studies, nov 2007, paper 2, question 1A

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baiermathstudies.wikispaces.com · Web view2012-03-08 · 1. Topics 3.1 & 3.3 . Set Theory. ... Often probability questions can be answered by drawing a Venn diagram to represent