EDEXCEL STATISTICS 1 PROBABILITY Basic Laws and Notation


 Job Green
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1 Objectives: Remember your basics from GCSE Develop your notation Introduce the addition formula Consider a new method Venn Diagrams You will find that you quickly extend your basic ideas on probability that you learned at GCSE level. Many questions still rely on identifying the sample space for a given event and the idea of equally likely outcomes. For some event A, the probability of it occurring lies between the two extremes, impossible 0 and certain 1. 0 PA ( ) 1 An event either happens or it doesn t! P( A¹) 1 P( A ) Sometimes the sample space is obvious to us, for example rolling a dice, picking from a standard pack of cards etc. At other times a simple list or table can help make things easier to see. Example A coin is thrown three times in a row what is the probability of getting Exactly one head? (b) At least two heads? There are several ways to deal with this but a simple list should sort it out for us. HHH THH HTH HHT TTT HTT THT TTH Clearly P (Exactly one head) 3 8 (b) P(At least one head) 1 P(No heads) Page 1 of 6
2 The moral of the story here, if the sample space is not obvious a simple list of equally likely outcomes might be all that is needed. Example Two fair dice are thrown what is the probability of getting A double (b) A score of at least 9 (c) A double or a score of at least 9 Again here, simple GCSE techniques will do us fine, a grid or table is all that is required to show the sample space and help highlight the solutions. 6 5 DICE DICE 1 6 Use the diagram to highlight the required outcomes and to confirm that P (A double) 6 36 (b) P (Score at least 9) (c) P (Double or score at least 9) = 36 Now you probably got part (c) by common sense but in fact it s a good example of an important probability law or axiom that you will need quite a bit. The Addition Law says that given 2 events A, B P( A or B) P( A) P( B) P( A and B ) Page 2 of 6
3 Let s check that Let A = Score a double B = Score at least 9 P( A or B) P( A) P( B) P( A and B) Here s the problem too many students at GCSE simply learn that OR means ADD, so why does it go wrong here? GCSE questions tend to ask you for two separate events that cannot occur together, picking the King of Spades or the King of Hearts etc Its easy to see here that there are scores of at least 9 that are also doubles (5,5) (6,6) and if we do not apply the addition law correctly these get counted twice. More formally GCSE questions tend to deal with mutually exclusive events. If two events A, B are such that PA ( and B) 0, ie they cannot both occur at the same time then they are said to be mutually exclusive. Then and only then!! does the addition law simplify to P( A or B) P( A) P( B ) Questions like the last one lend themselves nicely to another type of representation Venn Diagrams. Here is a very common type of problem which we can illustrate using this method. 100 people were surveyed about which Sunday newspaper they read, 50 people said the Times, 27 said The Observer and 11 said that they read both. We can represent this information as follows T O Page 3 of 6
4 Take a minute to make sure that you appreciate that 50 people read the Times is not the same as 50 people read only the times which you should see now from the Venn Diagram is 39 people. We can now use the diagram to help answer questions such as: What is the probability that a person chosen at random from the sample (b) Reads the Times or The Observer Reads the Observer but not The Times Again, you will probably be able to solve these by common sense, but they also give us the chance to develop our use of set notation and introduce another useful rule. Always define the events using commonsense notation Let T = Reads The Times O= Reads The Observer Reads the Times or The Observer P( T O) P( T) P( O) P( T O) Notice we have started to use the union symbol And the intersection symbol to mean and to mean or Key Point: The use of or means the first event or the second event or both together (b) Reads the Observer but not The Times. Now if you say this to yourself in words more clearly Reads the Observer and does not read The Times Using our new set notation this becomes P( T O) P( T) P( O) P( T O) Page 4 of 6
5 Some exam style questions take a purely abstract approach. Worked Example For the events A and B, P (A ) = 0.4 P( B ) = 0.3 and P (A B ) = 0.2 Find the value of P( A B ¹) (b) the value of P( A B ) (c) the value of P( A¹ B ¹) (d) the probability that only one of A and B occurs (2) (3) (2) (3) (Total 10 marks) Here a mixture of a Venn Diagram and our probability laws using set notation can help solve it A B P( A B¹) P( A) P( A B ) (b) P( A B) P( A) P( B) P( A B ) Page 5 of 6
6 (c) P( A¹ B¹) P( A B)¹ 1 P( A B) (d) the probability that only one of A and B occurs P( A B¹) P( B A ¹) For many questions, once the Venn diagram has been drawn and labelled correctly, they can just be answered by common sense. However, algebraically, the following laws are the most useful. P( A B) P( A) P( B) P( A B) P( A) P( A B) P( A B¹) P( A¹ B¹) P( A B)¹ 1 P( A B) Page 6 of 6
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