Transcript Document

The British actor Anthony Hopkins was delighted to
hear that he had landed a leading role in a film based
on the book The Girl From Petrovka by George Feifer.
A few days after signing the contract, Hopkins travelled
to London to buy a copy of the book. He tried several
bookshops, but there wasn't one to be had. Waiting at
Leicester Square underground for his train home, he
noticed a book apparently discarded on a bench.
Incredibly, it was The Girl From Petrovka. That in itself
would have been coincidence enough but in fact it was
merely the beginning of an extraordinary chain of
events.
Two years later, in the middle of filming in Vienna,
Hopkins was visited by George Feifer, the author.
Feifer mentioned that he did not have a copy of his
own book. He had lent the last one - containing his
own annotations - to a friend who had lost it
somewhere in London. With mounting astonishment,
Hopkins handed Feifer the book he had found. 'Is this
the one?' he asked, 'with the notes scribbled in the
margins?' It was the same book.
Win if you get a RED!
G
B
Would you play this game?
What is the probability of a red?
P(red) = 2/8 = 1/4
What is the probability of not
getting a red?
G
R
B
B
B
R
P(not a red) = 1 – P(red)
=1–¼=¾
What is the probability of
getting a red or a green?
P(r or a g) = P(red) + P(green)
= 2/8 + 2/8 = 4 /8 = 1/2
Lesson Objective
Understand that probability is a measure of how likely something is
to happen and be able to calculate probabilities for a single event
Know that P(something does not happen) = 1 – P(It does happen)
When a situation has several equally likely outcomes it is possible
to calculate the probability of an outcome occurring by using the
formula:
Probability (event) =
No. of ways an event can happen
Total number of all possible outcomes
This will give a value between 0 and 1, where 0 is
impossible and 1 is certain. Probability can either be
expressed as a fraction, decimal or a percentage.
We will use FRACTIONS, occasionally decimals.
Impossible
0
Unlikely
Even Chance
Likely
¼
½
¾
Certain
1
1a) ½
b) ½
c) ¼
d) 3/8
What if we have a more complicated situation?
G
B
G
R
B
B
B
R
I Spin the spinner
twice and I only win if
I get the exactly the
same colour on both
spins.
Spin 2
G
R
B
B
G
B
B R
R
R
G
G
B
B
B
B
R
RR
RR
RG
RG
RB
RB
RB
RB
R
RR
RR
RG
RG
RB
RB
RB
RB
G
RG
RG
GG
GG
GB
GB
GB
GB
G
RG
RG
GG
GG
GB
GB
GB
GB
B
RB
RB
BG
BG
BB
BB
BB
BB
B
RB
RB
BG
BG
BB
BB
BB
BB
B
RB
RB
BG
BG
BB
BB
BB
BB
B
RB
RB
BG
BG
BB
BB
BB
BB
Spin 1
Spin 2
G
R
B
B
G
R
B
B R
R
R
G
G
Spin 1
B
B
B
B
R
G
G
B
B
B
B
To calculate the probability of an event we need to consider all the
equally likely outcomes.
The list of equally likely outcomes is called the POSSIBILITY SPACE
Then we can use the formula:
Probability (event) =
No. of ways an event can happen
Total number of all possible outcomes
Lottery Card
Game
Lines
1
2
3
Choose 4 numbers.
The lotto numbers are going to be created by rolling 2
dice and adding the resulting total of each of the three
numbers together; so choose your numbers wisely.
The first to get a line wins the game.
Assuming that the dice really are random and fair, which numbers
were the best to choose and why?
Assuming that the dice really are random and fair, which numbers
were the best to choose and why?
There are 36 possible
combinations.
2nd die
P(2) = 1/36
P(4) = 3/36 = 1/12
etc
You can clearly see that
the best numbers to
chose are 5,6,7 and 8
or 6,7,8 and 9.
1st die
P(3) = 2/36 = 1/18
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10 11
6
7
8
9
10 11 12
Lesson Objective
Be able to use a Possibility Space Diagram to calculate the
probability of combined events
In my family there are two adults: Me, my partner Carol and my
two children Poppy and Lucy.
We sit in a line on a park bench to have a picnic.
Assuming that we sit down randomly, what is the probability that
the two children end up sitting next to each other?
In my family there are two adults: Me, my partner Carol and my
two children Poppy and Lucy.
We sit in a line on a park bench to have a picnic.
Assuming that we sit down randomly, what is the probability that
the two children end up sitting next to each other?
Suppose Granny Annie joins us and we have 5 people, what is the
probability that the children sit next to each other?
We need to draw up a list of the possible outcomes
(the possibility space)
MWLP
WMLP
LWPM
PWLM
MWPL
WMPL
LWMP
PWML
MLWP
WPML
LPMW
PMLW
MLPW
WPLM
LPWM
PMWL
MPLW
WLPM
LMPW
PLWM
MPWL
WLMP
LMWP
PLMW
There are 24 ways we can sit on the bench, so 24 possible
outcomes in the possibility space.
In 12 of them the children sit together so 12/24 = 1/2
When a situation has several equally likely outcomes it is possible
to calculate the probability of an outcome occurring by using the
formula:
Probability (event) =
No. of ways an event can happen
Total number of all possible outcomes
Eg When I flip 3 fair coins what is the probability that I get 3 Heads?
Two four-sided dice are thrown and the numbers added
together.
Construct a sample space diagram to show all the outcomes.
What is the probability of getting:
First die
Second die
+
1
2
3
4
1) a total more than 4?
1
2
3
4
5
2) a total less than 8?
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
3) a prime number total?
4) a total that is at least 3?
5) a total of 4 or 5?
6) the same number on both dice?
7) a lower number on the first dice?
Consider the following situation:
I roll two six sided die and look at the difference in the
scores. What is the probability that the difference in the
scores is a square number?
Lesson Objective
Understand when two events are mutually exclusive
Learn that when two events, A and B are mutually exclusive we can
use the formula P(A OR B) = P(A) + P(B)
Two events are mutually exclusive if they do not overlap
Eg
A = I pick a male
B = I pick a female
Eg
A = On a fair coin I flip a Head
B = On a fair coin I flip a Tail
Stand up if you think these ARE MUTUALLY EXCLUSIVE
On a fair die
A: Roll an even number
B: Roll an odd number
Stand up if you think these ARE MUTUALLY EXCLUSIVE
On a fair die
A: A randomly chosen word begins with the letter ‘a’
B: A randomly chosen word begins with the letter ‘b’
Stand up if you think these ARE MUTUALLY EXCLUSIVE
On a fair die
A: You roll a prime number
B: You roll an even number
Stand up if you think these ARE MUTUALLY EXCLUSIVE
When you look at a light
A: The light is on
B: The light is off
Stand up if you think these ARE MUTUALLY EXCLUSIVE
When you look outside
A: It is sunny
B: It is windy
Stand up if you think these ARE MUTUALLY EXCLUSIVE
A: You are male
B: You are in the top set
Write down your own example of a mutually
exclusive pair of events
First die
Second die
Use the table to find the probability
of getting a score of 3 or 4.
+
1
2
3
4
1
2
3
4
5
2
3
4
5
6
The probability of getting a score of
3 or 4 can be written as P(3 or 4).
3
4
5
6
7
Notice that:
4
5
6
7
8
P(3 or 4) = P(3) + P(4)
=
2
16
3
+ 16
=
5
16
So you can find this probability by simply adding the two separate
probabilities.
Similarly,
P(2 or 7) = P(2) + P(7) =
1
16
2
+ 16
=
3
16
Two six-sided dice are thrown.
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3 4
4 5
5 6
6 7
7 8
8 9
9 10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Work out P(3 or 4) by
adding fractions.
P(3 or 4) = P(3) + P(4)
P(5 or 6) = P(5) + P(6) =
=
2
36
3
+ 36
=
5
36
Work out P(5 or 6) by
adding fractions.
4
36
5
+ 36
=
9
36
=
1
4
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4 5 6
5 6 7
6 7 8
7 8 9
8 9 10
9 10 11
10 11 12
The probability of
getting an even total
when you roll two fair
dice is?
The probability of
getting a prime total
when you roll two fair
dice is?
So why isn’t:
18
P(a prime number OR an even number) =36
15
+36
33
=36
?
The OR Rule
People often use the fact that
P(A OR B) = P(A) + P(B)
But this is only true if the outcomes do not overlap,
Outcomes that do not overlap are called MUTUALLY
EXCLUSIVE
If they do overlap then this rule is no good because you
count the overlap twice – its better to count!
I roll a red die and a black die. Draw a possibility space diagram to
show the total scores for the two dice.
Find each probability and decide if the events are mutually
exclusive:
a) P(A total of 4 OR a total of 6)
b) P(A total of 8 OR a total that is prime)
c) P(A total that is even OR a total more than 9)
d) P(A 4 on the red die OR a 6 on the black die)
e) P(A total that is even OR a total that is prime)
f) P(A total that is more than 7 OR a 5 on the black die)
g) P(A total less than 5 OR a total more than 10)
I roll a red die and a black die.
Find each probability and decide if the events are mutually exclusive:
a) P(A total of 4 OR a total of 6)
= 8/36 ME
b) P(A total of 8 OR a total that is prime) = 20/36 ME
c) P(A total that is even OR a total more than 9) = 20/36
d) P(A 4 on the red die OR a 6 on the black die) = 11/36
e) P(A total that is even OR a total that is prime) = 32/36 ME
f) P(A total that is more than 7 OR a 5 on the black die) = 17/36
g) P(A total less than 5 OR a total more than 10) =9/36 ME
Lesson Objective
Consolidate our ability to find probabilities involving ‘OR’
Consolidate our use of the formula P(A OR B) = P(A) + P(B) for
mutually exclusive events and be able to adapt it for events that are
not mutually exclusive
Eg I roll a fair die labelled 1 to 6. What is the probability that:
a) I get an even number?
b) I get a square number?
c) Either a square number or an even number?
d) Is getting a square number a mutually exclusive event to getting
an even number?
Eg: 1/3 of the students in a room are in Year 7
¼ of the students in a room are in Year 8
If I pick a random student from the room what is the probability
that they are either from Year 7 or Year 8?
I roll a red die and a black die. Draw a possibility space diagram to
show the total scores for the two dice.
Find each probability and decide if the events are mutually
exclusive:
a) P(A total of 4 OR a total of 6)
b) P(A total of 8 OR a total that is prime)
c) P(A total that is even OR a total more than 9)
d) P(A 4 on the red die OR a 6 on the black die)
e) P(A total that is even OR a total that is prime)
f) P(A total that is more than 7 OR a 5 on the black die)
g) P(A total less than 5 OR a total more than 10)
I roll a red die and a black die.
Find each probability and decide if the events are mutually exclusive:
a) P(A total of 4 OR a total of 6)
= 8/36 ME
b) P(A total of 8 OR a total that is prime) = 20/36 ME
c) P(A total that is even OR a total more than 9) = 20/36
d) P(A 4 on the red die OR a 6 on the black die) = 11/36
e) P(A total that is even OR a total that is prime) = 32/36
f) P(A total that is more than 7 OR a 5 on the black die) = 17/36
g) P(A total less than 5 OR a total more than 10) =9/36 ME
1) The probability that I pick a red sweet from a bag is 0.3. The
probability that I pick a yellow sweet is 0.4. What is the probability
that a randomly chosen sweet is either red of yellow?
2) Seniors are competitors who are at least 65.
Adults are competitors between the ages of 16 and 21 inclusive.
Juniors are competitors under the age of 16.
1/
of the competitors are seniors and 1/3 are juniors.
If I randomly pick a competitor what is the probability that they are
in the adult category?
5
3) The probability that it rains on any given day is 0.2. The probability
that it is more than 12oC on any given day is 0.4. Why is the
probability that it either rains or is warmer than 12oC on any given
day not 0.6?
4) In a class of students 1/3 have blue eyes, ¼ have black hair and 1/8
have blue eyes and black hair. If I randomly pick a student at
random, what is the probability that they have either blue eyes or
black hair?
4) Mr B has a pack of cards, labelled from 1 to 50
If a student randomly picks a card, what is the probability :
a) that it is even?
b) that it is a multiple of 3?
c) that it is either even or a multiple of 3?
5) Mr B has a pack of cards, labelled from 1 to 100
If a student randomly picks a card, what is the probability :
a) that it is a multiple of 5?
b) that it is a multiple of 7?
c) that it is either a multiple of 5 or a multiple of 7?
6) In a class there are 30 students 10 have no pets. The rest of the
students either have a cat or a dog or both. 15 students say that they
have a dog and 11 students say that they have a cat. What is the
probability that a randomly chose student from the class has:
a) a pet
b) a pet dog
c) both a pet and a cat
d) either a pet cat or a pet dog or both
7) In a sixth form of 200 students, 60 do maths A level and 45 do
Biology A level. 12 study both Maths and Biology. If a randomly
chosen person from the sixth form is chosen what is the probability
that: a) They study both Maths and Biology
b) They study only maths
c) Are studying Maths and studying Biology mutually
exclusive?
8) Mr B has a pack of cards, labelled from 1 to 100
If a student randomly picks a card, what is the probability :
a) that it is a multiple of 3?
b) that it is a multiple of 5?
c) that it is either a multiple of 5 or a multiple of 7 but not an even
number?
9) Make up a question of your own where the probabilities are not
mutually exclusive and another question where the probabilities are
mutually exclusive.
Socks!
Four pairs of socks are
jumbled up in a drawer. If
you put your hand in without
looking, how many socks
must you take out to be
certain of getting a matching
pair?
What if there were 5 pairs?
…. 6?
Generalise!
Lesson Objective:
Be able to find the probability of one thing being followed by
another.
Begin to understand the difference between P(A followed by B)
and P(A and B)
Notice:
Names and events used in this lesson are random and have no basis in fact. Any
resemblance made to actual persons, living or dead, in the class room or outside
are purely coincidental.
A student in year 10 has 7 pairs of socks. However, being
disorganised they simply grab a pair of random socks out of the
draw at the start of each day. Unfortunately, they never get round to
washing their dirty socks and simply return them to the draw at the
end of each day.
The draw contains 5 pairs of black socks and 2 pairs of white (?!?)
socks.
a) What is the probability that the student wears white socks
on any particular day?
b) What is the probability that the student wears black socks
for two consecutive days?
c) What is the probability that the student wears black socks
on Monday followed by white socks on Tuesday, followed
by black socks on Wednesday?
In general:
P(A followed by B followed by C …..)
= P(A) × P(B) × P(C) …………
Eg On any given day the probability that a bus is late is 1/3
a) Find the probability that the bus is late two days running.
b) Find the probability that the bus is late for three
consecutive days.
c) Find the probability that from Monday to Friday the bus
is late on just Wednesday.
d) Find the probability that the bus is not late on the first of
three consecutive days, but is late on the other two.
In general:
P(A followed by B followed by C …..)
= P(A) × P(B) × P(C) …………
Eg In a bag there are twenty sweets. 12 are red. I pick a number of
sweets randomly from the bag without replacement. Find the
probability that:
a) I get a red sweet followed by a red sweet
b) I get 3 red sweets in a row
c) I get a red sweet followed by a different colour followed by a
red sweet.
1) The probability that it rains on a any given day is fixed at 1/5.
What is the probability that a) It rains on 3 successive days?
b) The second day of a weekend only?
2) A bag contains 3 red sweets, 2 blue sweets and a yellow sweet.
I draw three sweets from the bag. What is the probability that:
a) I get all red sweets?
b) I get a red sweet followed by a blue sweet followed by a yellow
sweet?
c) I get 3 blue sweets?
3) A class contains 7 male and 6 female students.
I randomly select 3 students. What is the probability that:
a) I select a male followed by a male followed by a female
b) I select a male followed by a female followed by a male
c) I select a female followed by a male followed by a male
d) What is the probability that one of the three is female?
I spin a coin three times
What is the probability that it will show heads, then tails, then
heads?
On any given day the probability that it rains is 1/3
What is the probability that it rains three days in a row
On any given day the probability that it rains is 1/3
What is the probability that it rains on only the first day of a
four day period
A bag contains 5 red balls and 3 green balls
I take 2 balls out of the bag at the same time
What is the probability that both balls are red?
A bag contains 5 red balls and 3 green balls
I take 3 balls out of the bag at the same time
What is the probability that the first ball is red, the second
green and the third red?
A bag contains 5 red balls and 3 green balls
I take 3 balls out of the bag at the same time
What is the probability that the first ball is red, the second
green and the third red?
I take out two cards from a pack of 52 without replacing them
What is the chance that I draw
A queen and then a king?
Plenary
In general:
P(A followed by B followed by C …..)
= P(A) × P(B) × P(C) …………
Eg The probability that I get a head when I flip a biased coin is 1/3
I flip the coin three times
a) What is the probability that I get 3 heads?
b) What is the probability that I get a head followed by two tails?
c) What is the probability that I get a single head?
Note the
difference!!!
Lesson Objective:
Consolidate out understanding of how to find the P(A and B) and
how it relates to P(A followed by B)
I roll a fair die three times, recording the score each time.
Find the probability that I get 3 sixes in a row?
What is the probability that I get a six on the first roll only?
What is the probability that I get a single six from the three rolls?
I roll a fair die three times, recording the score each time.
Find the probability that I get 3 sixes in a row?
What is the probability that I get a six on the first roll only?
What is the probability that I get a single six from the three rolls?
I roll three fair dice.
Find the probability that I get 3 sixes?
What is the probability that I get a six on only one die?
What is the probability that I get a single six?
A class has 10 girls and 8 boys. I pick two random names from the
class. What is the probability that I get:
a) Two girls
b) One girl
c) At least one girl
Worksheet Exercise 23c
Plenary: (This is an A* GCSE maths question)
Consider the following question:
My counter is on square number 1.
I spin a fair spinner numbered form 1 to 3 and move forward the
number of squares stated.
If I land on a black square I am out.
What is the probability that I will be out of the game at some point
in the next two goes.
1
2
3
4
5
Lesson Objective
Be able to use probability tree diagrams to answer probability
questions involving more than one event.
In a draw there are 4 blue socks and 2 red blocks.
If you take two socks from the draw without replacing them.
What is the probability that both socks are the same colour?
In a draw there are 4 blue socks and 2 red blocks.
If you take two socks from the draw without replacing them, draw a
tree diagram to show the possible outcomes. What is the probability
that both socks are the same colour?
Being a generous and warm hearted maths teacher, I decide to give
away two ‘free’ lunch-time detentions to the students in my year 10
maths class. To play fair no student can be given both detentions.
What is the probability that the detentions are split between the
sexes?
5 a 1/10 b 3/10 c 3/5
6 a 0.94 b 0.012 c 0.048
You have an equal chance
of going in any ‘forwards’
direction at a Junction.
Draw a tree
diagram to
show the maze.
What is the
probability that
you are eaten?
Lesson Objective
Calculating the probability that something happens at least once
Eg
I toss a biased coin three times, what is the probability that I get
at least one Head, if the probability of getting a head each time is
1/ ?
3
Lesson Objective
Calculating the probability that something happens at least once
Eg
I toss a biased coin three times, what is the probability that I get
at least one Head, if the probability of getting a head each time is
1/ ?
3
1/
H
H
=
9
H
H
T
Answer = 5/9
T
H T = 2/9
H
T H = 2/9
T
T T = 4/9
In general:
P(Something happens at least once)
= 1 – P(It doesn’t happen at all)
1) What is the probability that I roll at least one six when I
roll a fair dice twice?
2) What is the probability that I get at least one head when I
toss a fair coin three times?
3) If a bus is late with probability 1/3, what is the probability
that it is late: a) At least once in two days
b) At least once in five days
4) In a bag of sweets, 10 are red and 10 are blue. I take 3
sweets without replacing, what is the probability that I get
at least one red sweet?
5) In a class of 15 pupils what is the probability that at least
one student has the same birthday as someone else in
the group?
What is the probability in a class of 31 of us that at least two of us
have the same Birthday?
What is the probability in a class of 31 of us that at least two of us
have the same Birthday?
It is much easier to answer this question by calculating the probability
that no-one has the same Birthday as anyone and then do:
P(At least two of us have the same Birthday)
= 1 – P(no-one has the same Birthday)
Consider me. My Birthday is 31st of July – same day has Harry Potter!
Plenary: (This is an A* GCSE maths question)
Consider the following question:
My counter is on square number 1.
I spin a fair spinner numbered form 1 to 3 and move forward the
number of squares stated.
If I land on a black square I am out.
What is the probability that I will be out of the game at some point
in the next two goes.
1
2
3
4
5
Lesson Objective - ADDITIONAL MATHS EXAM ONLY!!
Be able to calculate probability of a certain number of successes
when an event is repeated lots of times with a fixed probability
of success
Suppose I roll a fair six sided die 4 times and count the number of
sixes that I get. What outcomes can I get?
What is the probability of each outcome?
Suppose I roll a fair six sided die 4 times and count the number of
sixes that I get. What outcomes can I get?
What is the probability of each outcome?
0
1
2
3
4
Sixes
Six
Sixes
Sixes
Sixes
Suppose I roll a fair six sided die 4 times and count the number of
sixes that I get. What outcomes can I get?
What is the probability of each outcome?
0
1
2
3
4
Sixes
Six
Sixes
Sixes
Sixes
Suppose the die was biased so that the probability of getting a six
is actually 1/3 how would this affect the probabilities?
Suppose I roll a fair six sided die 4 times and count the number of
sixes that I get. What outcomes can I get?
What is the probability of each outcome?
0
1
2
3
4
Sixes
Six
Sixes
Sixes
Sixes
What if I kept the probability of getting a six as 1/3 but rolled the
die six times. How would this alter my calculations? Which bits
would be easy to alter? Which bits would be harder top alter?
In general when a trial is repeated several times and the probability of
success each time remains fixed there are obvious patterns to the
method we can use to calculate the probabilities of a given number of
successes.
Eg The probability that a bus is late on any given day is fixed at 1/4
The bus is monitored over a 4 day period.
a) What is the probability that the bus is late on all 4 days?
b) What is the probability that the bus is late on exactly 1 day?
Suppose we monitor the bus over a 7 day period.
c) What is the probability that the bus is late on all 7 days?
d) What is the probability that the bus is late on exactly 1 day?
e) What is the probability that the bus is late on exactly 3 days?
When we can calculate probabilities in this way we are said to be
modelling the situation as a BINOMIAL DISTRIBUTION
The Binomial Distribution: Characteristics
You have ‘n’ trials.
Each independent and each with a two outcomes, success and
failure.
The probability of success ‘p’ remains fixed for each trial.
Then if we define X = the number of success in the ‘n’ trials
We can say that X~B(n , p)
P(getting 2 successes) =
P(getting 5 successes) =
P(getting ‘r’ successes) =
1) I roll a fair die 5 times. What is the probability that I get 3 sixes?
1) A biased coin has the probability of getting a head as 1/3. I toss
the coin 5 times, what is the probability that I get 3 heads?
3) A factory produces ‘widgets’. The probability that a widget is
faulty is 10%. If I check 6 widgets what is the probability that 4
are faulty?
4) When I roll a fair die the what is the probability that I get a score
less than 3? If I roll a fair coin 8 times what is the probability
that I get a score of less than on four occasions?
5) If X~B(10,0.25),
Find a) P(X = 2)
c) P(X<4)
b) P(X = 0)
d) P(X>8)
The Binomial Distribution: Characteristics
You have ‘n’ trials.
Each independent and each with a two outcomes, success and
failure.
The probability of success ‘p’ remains fixed for each trial.
Then if we define X = the number of success in the ‘n’ trials
We can say that X~B(n,p)
P(X=r) = nCr pr (1-p)n-r
Lesson Objective - ADDITIONAL MATHS EXAM ONLY!!
Be able to recognise when a situation can be modelled with a
Binomial distribution.
Be able to use the correct notation to describe a Binomial
distribution
Be able to calculate probabilities using a Binomial distribution
Consider the following situation.
I bag contains 5 red and 10 yellow balls.
I pick a ball at random look at it and replace it.
I repeat this experiment 6 times and count the number of yellow balls.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
I roll a fair die 4 times and count the number of sixes.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
I roll a fair die 4 times and count the number of times I get an even
score.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
A bag contains 4 red and 3 blue balls. I take 3 balls from the bag
without replacing them and count the number of times I get a red ball.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
I roll 5 biased die each with the probability of getting a six fixed at ¼
and I count the number of sixes.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
I roll a two fair die 8 times and count the number of times I get a total
score of 12.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
I randomly pick 4 different pupils from a classroom and count the
number of girls that are picked.
Decide whether the situation described represents a Binomial
distribution. If so describe the situation using the correct notation.
In the National lottery 6 balls are chosen from 49 balls numbered 1 to
49. I count the number of balls that have a score less than 10.
Can you think of an example of situation where the distribution
might be distributed as follows
X ~ B(20, 0.5) ??
The probability t hat a pen drawn at random from a large box of
pens is defective is 0.1. A sample of eight pens is taken. Find
the probability that it contains:
a) No defective pens
b) One defective pen
c) At least two defective pens
(Page 66 book)
A multiple-choice test consists of ten questions with four
answers for each, only one of which is correct. A student
guesses at the answers. Find the probability that he gets:
a) None correct
b) At least 1 correct
c) To pass he must get at least 8 correct, what is the probability
that he passes?
(Page 67 book)
Ex 4B page 68 book
Extensive research has shown that 1 person in every 4
is allergic to a particular grass seed. A group of 20
university students volunteer to try out a new treatment.
a) What is the expected number of allergic people in the
group?
b) What is the probability that exactly two people in the
group are allergic?
c) What is the probability that no more than two people in
the group are allergic?
d) How large a sample would be needed for the probability
of it containing at least one allergic person to be greater
than 99.9%.
1) A circuit board has 5 components. It will fail to work if
at least 3 of the components are faulty. If the
probability of a faulty component is 3/8. What is the
probability that any given circuit board is will not
work?
If you buy a box of 10 circuit boards. What is the
likelihood that more than 1 of the circuit boards in the
box is faulty?
Starter question:
What is the probability that it will snow on
Christmas day this year?
How might you try and assign a probability to
this event?
Lesson Objective
Understand that some situations are too difficult to model
using equally likely outcomes so probabilities need to be
found using an alternative technique
Understand how we can estimate the probability of an event
using relative frequency
The
Relative
=
Number of times the event occurs in
the experiment
Frequency
of an event
The total number of trials in the
experiment
Eg I check the weather every day in April.
It rains on 8 of the days, what is the relative frequency
of it raining in April?
Eg Records suggest that the relative frequency of a bus
being late in the morning is 0.1
Over a term of 34 days, on how many days would I
expect the bus to be late?
a) How many of Charlie’s first 50 arrows hit the target? (2 marks)
b) How many of Charlie’s arrows hit the target in the fifth week? (2 mks)
c) Estimate the probability that one of Charlie’s arrows hits the target.
The Monty Hall Problem
MCPT Mathematics and Technology
About Let’s Make a Deal
• Let’s Make a Deal was a game show hosted by
Monty Hall and Carol Merril. It originally ran
from 1963 to 1977 on network TV.
• The highlight of the show was the “Big Deal,”
where contestants would trade previous
winnings for the chance to choose one of three
doors and take whatever was behind it-maybe a car, maybe livestock.
• Let’s Make a Deal inspired a probability
problem that can confuse and anger the best
mathematicians.
Suppose you’re a contestant on Let’s Make a
Deal.
You are asked to choose one of three doors.
The grand prize is behind one of the doors;
The other doors hide silly consolation gifts
which Monty called “zonks”.
You choose a door.
Monty, who knows what’s behind each of the doors,
reveals a zonk behind one of the other doors.
He then gives you the option of switching doors or
sticking with your original choice.
You choose a door.
Monty, who knows what’s behind each of the doors,
reveals a zonk behind one of the other doors.
He then gives you the option of switching doors or
sticking with your original choice.
The question is: should you switch?
True or Not?
At the start of the game there is a 1/3 chance of me
picking the car.
I now know one of the doors which has a zonk
behind it, so there are two doors left, one of which
has the car and one of which has a zonk.
Therefore the chances of me winning the car is now
1/ for either door.
2
Conclusion: There is no point me changing my door
Is this true?
We are going to simulate the game in pairs.
One player in each pair will be Monty (the host)
and the other player will be the contestant
‘The Changers’ will play the game and always
change the door they select
‘The Stickers’ will play the game but never
change the door they select.
Each pair needs to play the game 10 times and
record how many wins
The correct answer:
You should change your choice, because the probability of you winning
the car if you do is 2/3.
You pick a door randomly
Pick a door
Pick a door
Pick a door
With a Zonk
With a Zonk
With a Car
Keep
Change
Keep
Change
Keep
Change
Win a
Win a
Win a
Win a
Win a
Win a
Zonk
Car
Zonk
Car
Car
Zonk
Plenary questions:
Why do we need to use relative frequency?
How do you calculate the relative frequency?
Do you think you will get better results by
calculating relative frequencies based on 10
experiments or 50 experiments?