SCNC1111 Scientific Method and Reasoning

Part IIb

Lecture 11 Probability

11-11-2014

The Additive Law of Probability Let and be any two events. We have

= + () Example: = , , ,, , = {,, ,}, = {,}, = , , ,, , ,

The Additive Law of Probability In the case with three events , , and , we have = + + +( )

Example 7 Students in a Science class are asked whether theyve ever visited the Hong Kong Disneyland (A), and whether theyve ever visited the Ocean Park (B). It is known that 75% of the students have visited the Hong Kong Disneyland, and 85% have visited the Ocean Park and 70% of the students have visited both. How many students have never visited both theme parks if there are 200 students in the class? It is known that = 0.75, = 0.85 and = 0.7. = + () = 0.75 + 0.85 0.7 = 0.9 = 1 0.9 = 0.1 The number of students who have never visited both theme parks is 0.1 200 = 20.

0.75 0.85 0.7

Example 8 (Venn diagrams) There are 50 first year students in the Department of Statistics and Actuarial Science. 50% of them intend to major in Statistics (A), 70% of them intend to major in Risk Management (B), and 20% of them intend to major in Actuarial Science (C). 15 intend to major in both A and B, 5 intend to major in both B and C, and 5 intend to major in both A and C. What is the probability that a randomly selected student intend to do a triple major in the department?

Cont. Example 8 Let , and denote the event of intending to major in Statistics, Risk Management and Actuarial Science, respectively. 1 = 0.5 + 0.7 + 0.2 1550 550 550 +

= 0.1

Conditional Probability The conditional probability of an event , given that an event has occurred is equal to

= ()

provided that () > 0.

Example 9 (Clinical Trial) In a study of a new treatment for insomnia, some randomly selected targeted patients were assigned to receive the tested drug for a year. Another group of randomly selected patients were assigned to receive a placebo treatment for a year. We check their responses 3 months after the treatment started in order to assess the long-term impact. The results of the clinical trial are shown in the table below:

What is the probability that a randomly selected patient received placebo treatment given that he had no relapse of insomnia? Let denotes the event that the patient received the placebo, and let denotes the event that the patient had no relapse. The require probability

= () = 3114071140 = 3171 = 0.4366

Response Treatment group Placebo Total Relapse 33 36 69 No Relapse 40 31 71 Total 73 67 140

Wikipedia, 2007.

Example 10 (Drawing balls) An urn contains 25 distinct numbered balls. Five balls are selected at random without replacement. Let = 2, 8, 18, 19, 21 , = { 18}. Find P . =

() = = 55 020

525 = 153130 = 0.0000188

= 11424525 = 1062653130 = 0.2

Hence, = () = 1531301

5

= 110626

= 0.0000941

The Multiplicative Law of Probability From the above definition, it is obvious that

= = (|) If A and B are independent, then = which is equivalent to stating that two events A and B are independent if = () or = (). Moreover, by the multiplicative law of probability, we have = = = (| ) or more generally, with k events 1, ,,

1 2 = 1 2 1 1 2 1 .

=

()

Example 11 Find the probabilities of randomly drawing 2 aces in succession from an ordinary deck of 52 playing cards (i) without replacement; (ii) with replacement.

(i) 452

351

= 1221

.

(ii) 452

452

= 1169

.

Birthday Problem n people in a room including you

dimensionsinfo, 2012.

Birthday Problem (A) = 1 = 1 () = 365

365 364

365 365+1

365

Hence = 1 365365

364365

365+1365

Birthday Problem (A)

Birthday Problem (B) =1 = 1 () = 365

365 364

365 364

365= 364

365

1

Hence q = 1 364365

1

Birthday Problem (B)

Birthday Problem (C) k No. of people required

1 23

2 14

3 11

4 9

5 8

6 8

7 7

8 7

SCNC1111Scientific Method and ReasoningPart IIbLecture 11ProbabilityThe Additive Law of ProbabilityThe Additive Law of ProbabilityExample 7Example 8Cont. Example 8Conditional ProbabilityExample 9Example 10The Multiplicative Law of ProbabilityExample 11Birthday ProblemBirthday Problem (A)Birthday Problem (A)Birthday Problem (B)Birthday Problem (B)Birthday Problem (C)