Win if you get a RED!

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.

BG

B

R B

G

RB

Win if you get a RED!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?

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.

CertainEven ChanceUnlikely Likely

0 1½¼ ¾

Impossible

1a) ½ b) ½ c) ¼ d) 3/8

What if we have a more complicated situation?

BG

B

R B

G

RB

I Spin the spinner twice and I only win if

I get the exactly the same colour on both

spins.

BG

B

R B

G

RB

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




Spin 1

Spin 2

BG

B

R B

G

RB

R R G G B B B B

R

R

G

G

B

B

B

B

Spin 1

Spin 2

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:


Lottery CardGame 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?

1211109876

111098765

10987654

9876543

8765432

7654321

654321+

2nd die

1s

t die

There are 36 possible combinations.

P(2) = 1/36

P(3) = 2/36 = 1/18

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.

Lesson ObjectiveBe 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?

Fun Maths

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?

Fun Maths

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)

There are 24 ways we can sit on the bench, so 24 possible outcomes in the possibility space.

MWLP

MWPL

MLWP

MLPW

MPLW

MPWL

WMLP

WMPL

WPML

WPLM

WLPM

WLMP

LWPM

LWMP

LPMW

LPWM

LMPW

LMWP

PWLM

PWML

PMLW

PMWL

PLWM

PLMW

MWLP

MWPL

MLWP

MLPW

MPLW

MPWL

WMLP

WMPL

WPML

WPLM

WLPM

WLMP

LWPM

LWMP

LPMW

LPWM

LMPW

LMWP

PWLM

PWML

PMLW

PMWL

PLWM

PLMW

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:


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.

What is the probability of getting:

1) a total more than 4?

2) a total less than 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?

87654

76543

65432

54321

4321+

Second die

Fir

st d

ie

Construct a sample space diagram to show all the outcomes.

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


On a fair die

A: A randomly chosen word begins with the letter ‘a’

B: A randomly chosen word begins with the letter ‘b’


On a fair die

A: You roll a prime number

B: You roll an even number


When you look at a light

A: The light is on

B: The light is off


When you look outside

A: It is sunny

B: It is windy


A: You are male

B: You are in the top set

Write down your own example of a mutually exclusive pair of events

87654

76543

65432

54321

4321+

Second dieF

irst

die

4

43

43

Notice that:

Use the table to find the probability of getting a score of 3 or 4.

The probability of getting a score of 3 or 4 can be written as P(3 or 4).

65

654

6543

543

7

7

2

65

65

65

5

P(3 or 4) = P(3) + P(4)

= + =216

316

516

So you can find this probability by simply adding the two separate probabilities.

316P(2 or 7) = P(2) + P(7) = + =

116

216

Similarly,

11

10

9

8

7

6

5

10

9

8

7

6

5

4 6321+

74321

129876

118765

107654

96543

85432

P(5 or 6) = P(5) + P(6) =

Two six-sided dice are thrown.

Work out P(3 or 4) by adding fractions.43

4

43

+ =236

336

536

P(3 or 4) = P(3) + P(4)

=

Work out P(5 or 6) by adding fractions.

6

6

5

6

65

65

5

43

4

43

+ =436

536

936 = 1

4

11

10

9

8

7

6

5

10

9

8

7

6

5

4 6321+

74321

129876

118765

107654

96543

85432

10

8

6

10

8

6

42

128

86

106

64

84

11

7

7

5 73

7

117

75

5

5

2

3

The probability of getting a prime total when you roll two fair dice is?

The probability of getting an even total when you roll two fair dice is?

So why isn’t:

P(a prime number OR an even number) = + = ?3336

1836

1536

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.


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.


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.


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.

1/5 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?

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!

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.

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)

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?

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.

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) × 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?



What is the probability that the first ball is red, the second green and the third red?



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) × 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 girlsb)One girlc)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 ObjectiveBe 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?

Calculating the probability that something happens at least once

Lesson Objective

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?

Calculating the probability that something happens at least once

Lesson Objective

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?

H

H

HT

T

T

H H = 1/9

H T = 2/9

T H = 2/9

T T = 4/9

Answer = 5/9

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?


0 Sixes1Six2Sixes3Sixes4Sixes



Suppose the die was biased so that the probability of getting a six is actually 1/3 how would this affect the probabilities?



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) b) P(X = 0)c) P(X<4) 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 distributionBe 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.


I roll a fair die 4 times and count the number of times I get an even score.


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.


I roll 5 biased die each with the probability of getting a six fixed at ¼ and I count the number of sixes.


I roll a two fair die 8 times and count the number of times I get a total score of 12.


I randomly pick 4 different pupils from a classroom and count the number of girls that are picked.


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

Frequency

of an event

Number of times the event occurs in the experiment

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.

The question is: should you switch?

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.

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

1/2 for either door.

Conclusion: There is no point me changing my doorIs 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

With a Zonk

Pick a door

With a Zonk

Pick a door

With a Car

Keep

Win a

Zonk

Keep

Win a

Zonk

Keep

Win a

Car

Change

Win a

Car

Change

Win a

Car

Change

Win a

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?

Documents

Win if you get a RED!