Trinity College
20th January 2014
Introduce..

The Fundamental Principle of
Counting
Operations
•The result of an operation is an outcome.
For example, if we throw a die, one
possible outcome is 5.
If we thr...
Suppose one operation has m possible
outcomes and that a second operation has n
outcomes. The number of possible outcomes
...
Activity: 

Example

draw two cards from a standard deck of 52 cards
without replacing the cards
     
There are 52 ways t...
Whats the Difference?

"The combination to
the safe was 472"

"My fruit salad
is a combination
of apples,
grapes and
banan...
We don't care what order the fruits are in, they could
also be "bananas, grapes and apples" or "grapes,
apples and bananas...
In Maths we use precise
language..
If the order doesn’t matter, it is a
COMBINATION
If the order does matter, it is a
PERM...
Permutations
or
Arrangements
Arrangements
Arrangements
Question 1
There are 6 different books, including a
science book, on a shelf.
In how many different ways can the six
books...
Solution
6! = 6 X 5 X 4 X 3 X 2 X 1= 720
1 X 5! = 1 X 5 X 4 X 3 X 2 X 1 = 120
Question 2
In how many ways can the letters of the
word SCOTLAND be arranged in a line?
In how many of theses arrangements...
Solutions
8! = 40320
7! X 2 = 10080 ways
5! x 2 = 240 ways
Question 3
How many different four digit numbers
greater than 6,000 can be formed using
the digits 1, 2, 3, 4, 5, 6, 7 if ...
Solution
(i) 2 X 5 X 4 X 3 = 120
(ii) 2 X 6 X 6 X 6 = 432
Combinations
n
r

The ( ) Notation
n

Can also be written as C aswell as nCr
r
and C(n,r).
It gives the number of ways of choosing r
ob...
How to Calculate It.
n) =
(r

n!
r! (n - r)!

Definition!

n) = n(n - 1)(n - 2)...(n - r +1)
(r
r!

Practical!
Important
n

(0)=1

n
(n) = 1
Question 1
In how many ways can a team of 5
players be chosen from 9 players?
In how many ways can this be done if a
certa...
Solution
9 ) = ( 9 ) = 9 X 8 X 7 X 6 = 126
(5
4
4X3X2X1

8
( 4 ) = 8 X 7 X 6 X 5 = 70
4X 3X2X1
The Twin Rule
n
n
It states that ( r = ( n-r
)
)

Proof:
LHS =
RHS =

=
= n!
(n - r)!(n - (n - r))!

=

n!
(n - r)!r!

= L...
Question 2
In how many ways can a committee of
six be formed from 5 teachers and 8
students if there are to be more teache...
Solution

Two possible combinations:

(i) 4 teachers and 2 students can be selected in
( 5 ) X ( 8) ways = 5 X 28 = 140 wa...
Equations using (n-c-r)
When you have to solve equations the following are
very useful.
n
( )=1
1

n
(2 ) = n(n - 1) = n(n...
Example
Solve for the value of the natural number
n
n such that ( ) = 28. 2
Solution

n(n - 1) = 28
2

n2 - n = 28
2
n2 - n = 28 -> n2 - n - 28 = 0
(n - 8)(n + 7) = 0
n=8

n=-7

Reject n = - 7 is no...
Find the value of n E N
in the following:
(n 2+ 1 ) = 28
SOL: (n + 1)(n) = 28
( ) = 28
2
n2 + n - 56 = 0
(n + 8)(n - 7) = ...
“What are the
chances of that
happening?”
Theory of Probability
17th Century Gambling
Chevalier de Mere gambled frequently to increase his
wealth.
He bet on a roll of a die that at least...
Correspondence leads to Theory
He asked his friend Blaise Pascal why this was the case.
Pascal worked it out and found tha...
Probability
is really about dealing with the unknown
in a systematic way, by scoping out the
most likely scenarios, or hav...
Terms
Definition

Example

An EXPERIMENT is a situation involving chance or
probability that leads to results called outco...
2013 Paper 2: Q 1
Binomial Distribution
or
Bernoulli Trials
Bernoulli Trials
Bernoulli Trials
Experiment that
satisfies:
A fixed number, n, of trials are carried out.
Each trial has two possible outcomes: success or
...
number of trials

probability of success
n
r

P ( r successes) = ( ) p q
probability of failure

r n-r
Question 1
An unbiased die is thrown 5 times. Find
the probability of obtaining
(i) 1 six
(ii) 3 sixes
(iii) at least 1 si...
Solution - Step 1
Let p = P(success i.e. getting a six) = 1/6.
Let q = P(failure i.e. NOT getting a six) = 5/6.
The number...
Solution - Step 2
n

P ( r successes) = ( r ) p q

r n-r

(i) P(1 six) = (
(ii) P( 3sixes)

5 1 4
1 )p q

5
1
4
)(1/6) (5/...
(iii) P(at least 1 six) = 1 - P(no six)
P(no six) = (

5 0 5
)p q
0

=(

5)(1/6)0(5/6)5
0

P(at least 1 six) = 1 - (5/6)
=...
You try: Question 2
Given that 10% of apples are bad, find the
probability that in a box containing 6
apples there is
(i) ...
Solution - Step 1
Let p = P(bad apple) = 1/10
Let q= P(good apple) = 9/10
Number of trials: n = 6
Solution - Step 2
6
0
6
(i) P(no bad apples) = ( 0 )(1/10) (9/10)

= 0.531441

(ii) P(1 bad apples) = (

6
1
5
)(1/10) (9/...
(iii) P(at least 1 bad apples) = 1 - P(no bad apples)
1 - (0.531441).....from

part (i)
Back to Question 1
We worked out the probability of getting
3 sixes wen a dice is thrown 5 times. If the
same dice is thro...
For a six to appear on the 10th throw,
(i) we need to get 3 sixes on the first nine
throws and then
(ii) get the 4th on th...
three sixes on the first nine throws is
given by:
9
3
6
( 3 ) (1/6) (5/6)

= 0.13

P(six on the 10th throw) = 1/6
Thus P(4...
Question 3
A card is drawn at random from a
normal deck of playing cards and then
replaced. The process is repeated until ...
Solution - Step 1
P(drawing a diamond is) = 1/4
The probability of drawing a third
diamond on the 10th draw is:
P(drawing ...
Solution - Step 2
P(2 diamonds on the 1st nine draws)
9
= ( )(1/4)2(3/4)7
2

= 0.3

P(diamond on the 10th draw) = 0.25
Thu...
Question 4
During match Sarah takes a number of penalty shots. The shots
are independent of each other and her probability...
Solution
Let p=P(she scores the penalty) = 0.8
Let q=P(she misses the shot) = 0.2
4 )(0.8)0(0.2)4 = 1/625
(i) (
0
4
(ii) (...
(iii)

n = 10

P(at least 8 scores) =
P(scores 8)
+ P(scores 9)
+ P(scores 10)
10
10
10 9
( )(0.8)8(0.2)2 + ( )(0.8) (0.2)...
Exam Questions
2012 Paper 2: Q 4
SOL:
Step 1

Let p=P(that she scores her free throw) = 0.6
Let q=P(she misses the free throw) = 0.4
The number of trials: ...
SOL:
Step 1

Step 2

P(scores once in the first 4 shots)
X
P(she scores on the 5th shot)
4
1
3
( )(0.6) (0.4)
1
0.1536

X
...
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Probability trinity college

  1. 1. Trinity College 20th January 2014
  2. 2. Introduce.. The Fundamental Principle of Counting
  3. 3. Operations •The result of an operation is an outcome. For example, if we throw a die, one possible outcome is 5. If we throw a die there are six possible outcomes: 1, 2, 3, 4, 5 or 6.
  4. 4. Suppose one operation has m possible outcomes and that a second operation has n outcomes. The number of possible outcomes when preforming the first operation followed by the second operation is m × n. Performing one operation and another operation means we multiply the number of possible outcomes. m×n
  5. 5. Activity:  Example draw two cards from a standard deck of 52 cards without replacing the cards       There are 52 ways to draw the first card.       There are 51 ways to draw the second card.       There are 52 X 51 = 2,652 ways to draw the two cards.
  6. 6. Whats the Difference? "The combination to the safe was 472" "My fruit salad is a combination of apples, grapes and bananas"
  7. 7. We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.
  8. 8. In Maths we use precise language.. If the order doesn’t matter, it is a COMBINATION If the order does matter, it is a PERMUTATION A PERMUTATION IS AN ORDERED COMBINATION.
  9. 9. Permutations or Arrangements Arrangements Arrangements
  10. 10. Question 1 There are 6 different books, including a science book, on a shelf. In how many different ways can the six books be arranged on a shelf? In how many ways can the 6 books be arranged if the science book is always on the extreme left?
  11. 11. Solution 6! = 6 X 5 X 4 X 3 X 2 X 1= 720 1 X 5! = 1 X 5 X 4 X 3 X 2 X 1 = 120
  12. 12. Question 2 In how many ways can the letters of the word SCOTLAND be arranged in a line? In how many of theses arrangements can two vowels come together? How many of these arrangements begin with S and end with the two vowels?
  13. 13. Solutions 8! = 40320 7! X 2 = 10080 ways 5! x 2 = 240 ways
  14. 14. Question 3 How many different four digit numbers greater than 6,000 can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if : (i) no digit can be repeated (ii) repetitions are allowed?
  15. 15. Solution (i) 2 X 5 X 4 X 3 = 120 (ii) 2 X 6 X 6 X 6 = 432
  16. 16. Combinations
  17. 17. n r The ( ) Notation n Can also be written as C aswell as nCr r and C(n,r). It gives the number of ways of choosing r objects from n different objects. It is pronounced ‘n-c-r’ or ‘n-choose-r’.
  18. 18. How to Calculate It. n) = (r n! r! (n - r)! Definition! n) = n(n - 1)(n - 2)...(n - r +1) (r r! Practical!
  19. 19. Important n (0)=1 n (n) = 1
  20. 20. Question 1 In how many ways can a team of 5 players be chosen from 9 players? In how many ways can this be done if a certain player must be selected in each team?
  21. 21. Solution 9 ) = ( 9 ) = 9 X 8 X 7 X 6 = 126 (5 4 4X3X2X1 8 ( 4 ) = 8 X 7 X 6 X 5 = 70 4X 3X2X1
  22. 22. The Twin Rule n n It states that ( r = ( n-r ) ) Proof: LHS = RHS = = = n! (n - r)!(n - (n - r))! = n! (n - r)!r! = LHS
  23. 23. Question 2 In how many ways can a committee of six be formed from 5 teachers and 8 students if there are to be more teachers than students on each committee?
  24. 24. Solution Two possible combinations: (i) 4 teachers and 2 students can be selected in ( 5 ) X ( 8) ways = 5 X 28 = 140 ways 4 2 (ii) 5 teachers and 1 student can be selected in ( 5 ) X ( 8 ) = 1 X 8 = 8 ways 3 1 The total number of possible committees is 140 + 8 = 148 committees
  25. 25. Equations using (n-c-r) When you have to solve equations the following are very useful. n ( )=1 1 n (2 ) = n(n - 1) = n(n - 1) 2×1 2
  26. 26. Example Solve for the value of the natural number n n such that ( ) = 28. 2
  27. 27. Solution n(n - 1) = 28 2 n2 - n = 28 2 n2 - n = 28 -> n2 - n - 28 = 0 (n - 8)(n + 7) = 0 n=8 n=-7 Reject n = - 7 is not a natural number. Therefore n = 8.
  28. 28. Find the value of n E N in the following: (n 2+ 1 ) = 28 SOL: (n + 1)(n) = 28 ( ) = 28 2 n2 + n - 56 = 0 (n + 8)(n - 7) = 0 n = -8 and n=7 Reject n = -8 an accept n = 7.
  29. 29. “What are the chances of that happening?” Theory of Probability
  30. 30. 17th Century Gambling Chevalier de Mere gambled frequently to increase his wealth. He bet on a roll of a die that at least one six would appear in four rolls. Tired of this approach he decided to change his bet to make it more challenging. He bet that he would get a total of 12, a double 6, on 24 rolls of two dice. The old method was most profitable!
  31. 31. Correspondence leads to Theory He asked his friend Blaise Pascal why this was the case. Pascal worked it out and found that the probability of winning was only 49.1% with the new method compared to 51.8% using the old approach! Pascal wrote a letter to Pierre De Fermat, who wrote back. They exchanged their mathematical principles and problems and are credited with the founding of probability theory.
  32. 32. Probability is really about dealing with the unknown in a systematic way, by scoping out the most likely scenarios, or having a backup plan in case those most likely scenarios don’t happen. Life is a sequence of unpredictable events, but probability can be used to help predict the likelihood of certain events occurring.
  33. 33. Terms Definition Example An EXPERIMENT is a situation involving chance or probability that leads to results called outcomes. What color would we land on? A TRIAL is the act of doing an experiment in P! Spinning the spinner. The set or list of all possible outcomes in a trial is called the SAMPLE SPACE. Possible outcomes are Green, Blue, Red and Yellow. An OUTCOME is one of the possible results of a trial. Green. An EVENT is the occurrence of one or more specific outcomes One event in this experiment is landing on blue.
  34. 34. 2013 Paper 2: Q 1
  35. 35. Binomial Distribution or Bernoulli Trials Bernoulli Trials Bernoulli Trials
  36. 36. Experiment that satisfies: A fixed number, n, of trials are carried out. Each trial has two possible outcomes: success or failure. The trials are independent. The probability of success in each trial is constant. The probability of a success is generally called p. The probability of a failure is q, where p + q = 1.
  37. 37. number of trials probability of success n r P ( r successes) = ( ) p q probability of failure r n-r
  38. 38. Question 1 An unbiased die is thrown 5 times. Find the probability of obtaining (i) 1 six (ii) 3 sixes (iii) at least 1 six.
  39. 39. Solution - Step 1 Let p = P(success i.e. getting a six) = 1/6. Let q = P(failure i.e. NOT getting a six) = 5/6. The number of trials: n = 5.
  40. 40. Solution - Step 2 n P ( r successes) = ( r ) p q r n-r (i) P(1 six) = ( (ii) P( 3sixes) 5 1 4 1 )p q 5 1 4 )(1/6) (5/6) 1 =( =3125/7776 5 3 2 = ( 3)p q =125/3888 =( 5 )(1/6)3(5/6)2 3
  41. 41. (iii) P(at least 1 six) = 1 - P(no six) P(no six) = ( 5 0 5 )p q 0 =( 5)(1/6)0(5/6)5 0 P(at least 1 six) = 1 - (5/6) = 4651/7776 5
  42. 42. You try: Question 2 Given that 10% of apples are bad, find the probability that in a box containing 6 apples there is (i) no bad apple (ii) just one bad apple (iii) at least one bad apple
  43. 43. Solution - Step 1 Let p = P(bad apple) = 1/10 Let q= P(good apple) = 9/10 Number of trials: n = 6
  44. 44. Solution - Step 2 6 0 6 (i) P(no bad apples) = ( 0 )(1/10) (9/10) = 0.531441 (ii) P(1 bad apples) = ( 6 1 5 )(1/10) (9/10) 1 = 0.015640313
  45. 45. (iii) P(at least 1 bad apples) = 1 - P(no bad apples) 1 - (0.531441).....from part (i)
  46. 46. Back to Question 1 We worked out the probability of getting 3 sixes wen a dice is thrown 5 times. If the same dice is thrown continuously until a six appears for the 4th time, how do we find the probability that the 4th six will appear on the 10th throw?
  47. 47. For a six to appear on the 10th throw, (i) we need to get 3 sixes on the first nine throws and then (ii) get the 4th on the 10th throw!
  48. 48. three sixes on the first nine throws is given by: 9 3 6 ( 3 ) (1/6) (5/6) = 0.13 P(six on the 10th throw) = 1/6 Thus P(4th six on the 10th throw) = 0.13 X 1/6 = 0.0217
  49. 49. Question 3 A card is drawn at random from a normal deck of playing cards and then replaced. The process is repeated until the third diamond is appears. Find the probability that this happens when the tenth card is drawn.
  50. 50. Solution - Step 1 P(drawing a diamond is) = 1/4 The probability of drawing a third diamond on the 10th draw is: P(drawing 2 diamonds in the first nine draws) X P(drawing a diamond on the 10th draw)
  51. 51. Solution - Step 2 P(2 diamonds on the 1st nine draws) 9 = ( )(1/4)2(3/4)7 2 = 0.3 P(diamond on the 10th draw) = 0.25 Thus P(third diamond on the 10th draw) = 0.25 X 0.3 = 0.075
  52. 52. Question 4 During match Sarah takes a number of penalty shots. The shots are independent of each other and her probability of scoring with each shot is 4/5. (i) Find the probability that Sarah misses each of her first four shots. (ii) Find the probability that Sarah scores exactly three of his first four penalty shots. (iii) It Sarah takes 10 penalty shots during the match, find the probability that she scores at least 8 of them.
  53. 53. Solution Let p=P(she scores the penalty) = 0.8 Let q=P(she misses the shot) = 0.2 4 )(0.8)0(0.2)4 = 1/625 (i) ( 0 4 (ii) ( 3 )(0.8)3(0.2)1 = 256/625
  54. 54. (iii) n = 10 P(at least 8 scores) = P(scores 8) + P(scores 9) + P(scores 10) 10 10 10 9 ( )(0.8)8(0.2)2 + ( )(0.8) (0.2)1 + ( )(0.8)10(0.2)0 = 10 8 9 ANS = 6,619,136 9,765,625
  55. 55. Exam Questions
  56. 56. 2012 Paper 2: Q 4
  57. 57. SOL: Step 1 Let p=P(that she scores her free throw) = 0.6 Let q=P(she misses the free throw) = 0.4 The number of trials: n = 6 Step 2 Is she scores four of the shoots she misses two! ( 6 )p4q2 = ( 6 )(0.6)4(0.4)2 4 4 = 972/3125 0.31104
  58. 58. SOL: Step 1 Step 2 P(scores once in the first 4 shots) X P(she scores on the 5th shot) 4 1 3 ( )(0.6) (0.4) 1 0.1536 X 0.6 = 0.09216

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