2.2 laws of probability (1)

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1. Laws of Probability 2.2

2. 2.2 Laws of probability - Targets Calculate conditional probability from a table. Apply the law P(A ∪ B) = P(A) + P(B) – P(A ∩ B ) to all events. Apply the law P(A ∩ B) = P(A)P(B I A) to events which are not independent. Solve probability problems using the laws of probability or tree diagrams. Stretch and challenge: How does a tree diagram with 2 outcomes on each branch link in with the binomial expansion/probability?

3. Conditional probability A room contains 25 people. The table below shows the numbers of each sex and whether or not they are wearing glasses. A person is selected at random. F is the event that the person selected is female. G is the event that the person selected is wearing glasses.

4. Conditional probability There are a total of 9 people wearing glasses so P(G) = . Only 4 of the nine males are wearing glasses so the probability of a male wearing glasses is while for the females the probability is . This means the probability of event G occurring is affected by whether or not event F has occurred. The 2 events are not independent .  

5. Conditional probability The conditional probability that the person selected is wearing glasses given that they are female is denoted P(G I F). P(A I B) denotes the probability that event A happens given that event B happens. Two events A and B are independent if: P(A) = P(A I B)

6. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: P(M) There are 40 male students out of a total of 70 students. = 0.571  

7. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: b ) P(R) 21 students are studying Russian. P(R) = = 0.3  

8. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: c) P(M I R) There are 21 students studying Russian and 14 are male. = 0.67  

9. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: d ) P(M ∩ R) There are 14 students who are both male and studying Russian = 0.2  

10. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: e ) P(M ∪ R) There are 17 + 9 + 14 + 7 = 47 students who are either male or studying Russian (or both). = 0.671  

11. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: f ) P(R I M) There are 40 male students of which 14 are studying Russian = 0.35  

12. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: g ) P(M ’ ) There are 30 students who are not male = 0.429  

13. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: h ) P(R ’ ) There are 29 + 20 = 49 students not studying Russian = 0.7  

14. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: i ) P(R I M ’ ) Of the 30 who are not male, 7 are studying Russian. = 0.233  

15. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: j ) P(R ’ I M) Of the 40 males, 17 + 9 = 26 are not studying Russian = 0.65  

16. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: k ) P(M ’ ∩ R) There are 7 students who are not male and are studying Russian, = 0.1  

17. Example 1 – 3.5 Page 44/45 Students on the first year of a science course at a university take an optional language module. The number of students of each sex choosing each available language is shown below. A student is selected at random M is the probability that the student selected is male R is the event that the student selected is studying Russian Write down the value of: l ) P(M ∪ R ’ ) There are 40 + 12 + 11 = 63 students who are either male or not studying Russian (or both), = 0.9.  

18. Addition law of probability In 2.1 we saw that if A and B are mutually exclusive events then P(A ∪ B) = P(A) + P(B ). A more general form of this expression which applies whether or not A and B are mutually exclusive events is P(A ∪ B) = P(A) + P(B) - P(A ∩ B) This is known as the addition law of probability

19. Addition law of probability Diagram on page 45 and subsequent working on 45/46 illustrates this Also see next example for how a numerical example proof

20. Example 2 150 students at a catering college have to choose between 3 tasks as part of their final assessment. The table on example 3.6 on page 46 gives their choices. A student is randomly selected. C denotes the event that the selected student chooses cake baking. D denotes the event that the selected student is female. Verify that P(C ∪ D) = P(C) + P(D) – P(C ∩ D)

21. Example 2 P(C ∪ D) = P(C) + P(D) – P(C ∩ D ) P(C) => There are 54 + 26 = 80 cake-baking students. P(C) = P(D) => There are 26 + 18 + 16 = 60 female students P(D) = P(C ∩ D ) => There are 26 female cake-baking students. P(C ∩ D ) = P(C) + P(D) – P(C ∩ D ) = + - P(C) + P(D) – P(C ∩ D) = There are 54 + 26 + 18 + 16 = 114 students who are either cake-baking or female or both P(C ∪ D) = = P(C) + P(D) – P(C ∩ D ) as required.  

22. Task 1 Exercise 3E Questions 1 and 3.

23. Task 1 Answers 1a 1b 1c 1d 1e 1f 1g 1h 1i 1j   2a 2 b 2 c 2 d 2 e 2 f 2 g 2 h 2 i   3a 3 b 3 c 3 d 3 e 3 f 3 g 3 h 3i 0 3j A and C not independent  

24. Multiplication law If A and B are independent events, then P(A ∩ B) = P(A)P(B). This is a special case of the multiplication law for probability. P(A ∩ B) = P(A)P(B I A )

25. Multiplication law Verify this by example A person is selected at random F is the event that the person is selected is female. G is the event that the person selected is wearing glasses. P(F ∩ G), the probability that the person selected is a female wearing glasses is or 0.2. P(F) = = 0.64 and P(G I F) = = 0.3125 P(F)P(G I F) = 0.2 P(F)P(G I F) = P(F ∩ G)  

26. Example 3 – 3.7 Page 49 Sheena buys ten apparently identical oranges. Unknown to her, two of the oranges are rotten. She selects two of the ten oranges at random and gives them to her grandson. Find the probability that: Both the oranges are rotten Exactly one of the oranges is rotten.

27. Example 4 – 3.8 Page 50 When Bali is on holiday she intends to go for a five-mile run before breakfast each day. However, sometimes she stays in bed instead. The probability that she will go for a run on the first morning is 0.7. Thereafter, the probability she will go for a run is 0.7 IF she went on the previous morning and 0.6 if she did not. Find the probability that on the first 3 days of the holiday she will go for: 3 runs Exactly 2 runs

28. Task 2 Exercise 3F Pages 50-52 Questions 1, 3, 5 and 6

29. Task 2 Answers 1a 0.195 1b 0.499 3ai 0.0152 3aii 0.182 3aiii 0.227 3bi 0.0909 3bii 0.136 3biii 0.409 3biv 0.218 5 a 0.196 5 b 0.0240 5c 0.0840 5d 0.0960 5e 0.240 5f 0.228 5g 0.192 6a 0.0461 6b 0.233 6 c 0.0121 6 d 0.279 6 e 0.0101 6 f 0.266 6 g 0.152

30. 2.2 Laws of probability – Key Points P(A I B) denotes the probability that event A happens given that event B happens. Two events A and B are independent if: P(A) = P(A I B) Addition law of probability P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Multiplication law of probability P(A ∩ B) = P(A)P(B I A )