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					COMP 10020


Lecturer: Dr. Saralees Nadarajah

Office: Room 2.223, Alan Turing Building

Email: mbbsssn2@machester.ac.uk
COMP 10020 Lectures


1.   Mondays 10am, Humanities Bridgeford
     Cordingley



2.   Thursdays 9am, Kilburn 1.1
COMP 10020 Example Classes


1.   Mondays 4pm, LF15
2.   Tuesdays 11am, LF15
3.   Tuesdays 2pm, LF15
4.   Fridays 12am, LF15
COMP 10020 Website




http://www.maths.man.ac.uk/service/COMP10020/materials.php
COMP 10020 Requirements

Co-requisites:    None



Pre-requisites:   None



Textbook: Not required
COMP 10020 Topics

1.   Axioms of probability

2.   Conditional probability and independence

3.   Random variables

4.   Some discrete distributions.
COMP 10020 Assessment


1.   Two take-home assignments



2.   Two questions in the final exam
Sets



       Sample Space
Sets



       Ec
            E
Sets




                   E         F




       E∩F is the overlap area (∩ =AND)
Sets




                 E        F




   EUF is the area Red or Yellow (U=OR)
Sets




              E         F




       E∩FC is the area in Red
Sets




                E         F




       EC∩F is the area in Yellow
Sets




             E               F




       E and F are mutually exclusive
Probability



           No of outcomes for E
Pr (E) =
            Total no of outcomes
Example 1

A hat contains four slips numbered 1 to 4.
Two drawn without replacement:

Sample Space = ?
Example 2

Toss a coin:

Sample Space = ?

Pr(Head) = ?

Pr(Tail) = ?
Example 3

Toss a 6-sided dice:

Sample Space = ?

Pr(“1” turned up) = ?

Pr(“2” turned up) = ?
Example 4

A card picked from ordinary bridge deck:

Pr(Card is ♠) = ?

Pr(Card is King) = ?
Example 5
Toss three coins:

Sample Space = ?

Pr(No heads) = ?

Pr(At least one head) = ?
Axioms of Probability


   0≤Pr(E)≤ 1

   Pr(Sample Space) = 1

   Pr(EUF) = Pr(E) + Pr (F) if E and F are
    mutually exclusive
Additive Law




    Pr(EUF) = Pr(E) + Pr (F) – Pr(E∩F)
Complementary Law 1




          Pr(EC) = 1 - Pr(E)
Complementary Law 2




      Pr(EC∩F) = Pr(F) – Pr(E∩F)
Complementary Law 3




      Pr(E∩FC) = Pr(E) – Pr(E∩F)
Example 6

Given: Pr(E)=0.4, Pr(F)=0.3, Pr(E∩F)=0.2.

Pr(EUF) = ?

Pr(EC) = ?

Pr(Fc) = ?
Assignment 3



   Due 2 December 2009 (Wednesday)



   Hand in to the Support Office
Example 7

A bowl contains slips numbered 1,2,…, 20.
A slip drawn at random and its number noted:

Sample Space = ?

Pr(Number is Prime OR Divisible by 3) = ?
Example 8

The probability that a student passes
  mathematics is 2/3, and the probability that
  he passes biology is 4/9.If the probability of
  passing at least one course is 4/5, what is the
  probability that he will pass both courses?
Example 9

A bag contains 5 balls, 3 are red and 2 are
  yellow. Three balls are drawn without
  replacement. Describe the sample space.
Example 10

Let C be the event “exactly one of the events A
  and B occurs.” Express Pr (C) in terms of Pr
  (A), Pr (B) and Pr (A ∩ B).
Example 11

A six-sided die is loaded in a way that each
  even face is twice as likely as each odd face.
  All even faces are equally likely, as are all
  odd faces. For a single roll of this die find the
  probability that the outcome is less than 4.
Example 12

Is the following statement true: if A and B are
  mutually exclusive events then Pr (A ∩ B) =
  Pr (A) Pr (B). Justify your answer with a
  simple example.
Example 13

Anne and Bob each have a deck of playing
  cards. Each flips over a randomly selected
  card. Assume that all pairs of cards are
  equally likely to be drawn. Determine the
  following probabilities:
(a) the probability that at least one card is an
  ace,
(b) the probability that the two cards are of the
  same suit,
(c) the probability that neither card is an ace.
Example 14


A die is rolled and a coin is tossed, find the
  probability that the die shows an odd number
  and the coin shows a head.
Example 15

A six--sided die is loaded in a way that each
  even face is twice as likely as each odd face.
  All even faces are equally likely, as are all
  odd faces. For a single roll of this die find the
  probability that the outcome is less than 4.
Example 16


Out of the students in a class, 60% are
 geniuses, 70% love chocolate, and 40% fall
 into both categories. Determine the
 probability that a randomly selected student
 is neither a genius nor a chocolate lover.
Example 17

Is the following statement true: if A and B are
  mutually exclusive events then Pr (A ∩ B) =
  Pr (A) Pr (B). Justify your answer with a
  simple example.
Example 18

If Pr (A) = 0.5 and Pr (B) = 0.4,but we have no
   further information about the events A and B,
   how big might Pr (A U B) be? How small
   might it be? How big might Pr (A ∩ B) be?
   How small might it be?
Example 19


Prove that for any two events A and B, we have
  Pr (A ∩ B)  Pr (A) + Pr (B) - 1.
Example 20

Let C be the event “exactly one of the events A
  and B occurs.” Express Pr (C) in terms of Pr
  (A), Pr (B) and Pr (A ∩ B).

				
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