Monty Hall’s problem is based on a game show, Let's make a deal, hosted by Monty Hall. In the show, the contestant has to choose a door out of three doors. Out of the three doors, there is one door containing a car and two containing goats. The contestant first chooses a door. The host, who knows what the door contains, will reveal another door with the goat. The contestant is then given a choice whether he wants to change or stay with the door he chose.
Since there are three doors to choose from, the initial choice is 1/3. After the contestant chooses one door, the chance of getting the car is still 1/3. After the host opens a door that contains a goat, the chances of getting the right door when you switch will become 2/3.
In all 3 Scenarios, the contestant chooses the first door.
Scenario 1: The first door has the car. The host reveals the door with a goat. If the contestant were to switch, he/she will get the goat instead of the car.
Scenario 2: The second door has the car. The host reveals the third door. If the contestant switches, he/she will get the car.
Scenario 3: The third door has the car. The host reveals the second door. If the contestant switches, he/she will get the car.
Hence, the chances are 2/3.
There are many misconceptions of the problem, for example, many people think that the choice is 1/3 and changed to 1/2 because there are only two doors left to choose from. However, as explained above, this is not the case.
Friday, August 19, 2011
Maths for Fun
Title: Fantastic Math Tricks
Resources Type: Website
Link: http://www.angelfire.com/me/marmalade/mathtips.html
Reason for recommendation: This is another website that teaches us how to calculate faster mentally. It is very useful for tests and competitions where calculators are allowed.
Resources Type: Website
Link: http://www.angelfire.com/me/marmalade/mathtips.html
Reason for recommendation: This is another website that teaches us how to calculate faster mentally. It is very useful for tests and competitions where calculators are allowed.
Maths for Fun
Title: Prime Factorisation Tool
Resource type: tool
URL: http://www.mathsisfun.com/numbers/prime-factorization-tool.html
Reasons for recommendation: This is also a very useful tool that we can use as some numbers can be hard to prime factorise, especially if they are big. So, you can use this tool to help you. Prime factorisation is useful in many situations such as when you need to find the LCM and HCF of certain numbers. Hence, I feel that everyone should know about this.
Resource type: tool
URL: http://www.mathsisfun.com/numbers/prime-factorization-tool.html
Reasons for recommendation: This is also a very useful tool that we can use as some numbers can be hard to prime factorise, especially if they are big. So, you can use this tool to help you. Prime factorisation is useful in many situations such as when you need to find the LCM and HCF of certain numbers. Hence, I feel that everyone should know about this.
Maths for Fun
Title: Learn how to calculate faster than the speeding calculator
Resource Type: Website
Link: http://www.glad2teach.co.uk/fast_maths_calculation_tricks.htm
Reasons for recommendation: This website teach us everything about mental calculation and if one were to memorise all the formulas in here, he/she will be very fast at mental calculation because this website has a lot of formulas and tricks for us to learn. This website is a must try for everybody.
Resource Type: Website
Link: http://www.glad2teach.co.uk/fast_maths_calculation_tricks.htm
Reasons for recommendation: This website teach us everything about mental calculation and if one were to memorise all the formulas in here, he/she will be very fast at mental calculation because this website has a lot of formulas and tricks for us to learn. This website is a must try for everybody.
Math for Fun
Title: Circle Music Video by Peter Weatherall
Resource type: Video
Link: http://www.youtube.com/watch?v=FbUPyUi5YtM&feature=channel_video_title
Reason for recommendation: This is quite an interesting song and I feel that you can learn basic things about circles from this video. It is formed in such a way that even a 9 year old will be able to understand it without having any prior knowledge of it. Hence, I want to share it.
Resource type: Video
Link: http://www.youtube.com/watch?v=FbUPyUi5YtM&feature=channel_video_title
Reason for recommendation: This is quite an interesting song and I feel that you can learn basic things about circles from this video. It is formed in such a way that even a 9 year old will be able to understand it without having any prior knowledge of it. Hence, I want to share it.
Math for Fun
Title: Fast Maths Trick of Calculation
Resource type: Video
Link: http://www.youtube.com/watch?v=hLdKsKep1og
Reason for recommendation: Even though this only works with multiples of number less than 20, it is very useful because we are too reliant on calculators many times. With this, calculation in Maths Olympiads will be much easier too and they can have fun conquering the "difficulties" of maths.
Resource type: Video
Link: http://www.youtube.com/watch?v=hLdKsKep1og
Reason for recommendation: Even though this only works with multiples of number less than 20, it is very useful because we are too reliant on calculators many times. With this, calculation in Maths Olympiads will be much easier too and they can have fun conquering the "difficulties" of maths.
Math for fun
Title: Fun Math Trick! Squaring Numbers Near 100
Resource: Video
Link: http://www.youtube.com/watch?v=JKWpI2EuIDA
Reasons for recommendation: This video definitely speeds things up for squaring numbers near 100, especially when we don't have a calculator. This can be useful in times of Maths Olympiads where calculators are not allowed. Even though it may seem difficult, it is actually very easy with this method. Hence, more people should know about this.
Resource: Video
Link: http://www.youtube.com/watch?v=JKWpI2EuIDA
Reasons for recommendation: This video definitely speeds things up for squaring numbers near 100, especially when we don't have a calculator. This can be useful in times of Maths Olympiads where calculators are not allowed. Even though it may seem difficult, it is actually very easy with this method. Hence, more people should know about this.
Lesson Reflections on Pythagorean Triplets
How did you feel about this lesson?
I feel that this lesson is one of the most interesting lessons of the year. Firstly, this is because I have always wanted to know formulas for Pythagoras' Theory because many Math Olympiads test on that. Secondly, this helps me in tests where I have no other option.
What have you learnt in this lesson?
I have learned the 3 theories that was taught in class by my classmates. I also know the basic ratios that can be used for right angled triangles.
Do you still have any doubts pertaining to this lesson?
I still am not very clear of the formulas and I feel that I should study more, or at least memorise the formulas.
I feel that this lesson is one of the most interesting lessons of the year. Firstly, this is because I have always wanted to know formulas for Pythagoras' Theory because many Math Olympiads test on that. Secondly, this helps me in tests where I have no other option.
What have you learnt in this lesson?
I have learned the 3 theories that was taught in class by my classmates. I also know the basic ratios that can be used for right angled triangles.
Do you still have any doubts pertaining to this lesson?
I still am not very clear of the formulas and I feel that I should study more, or at least memorise the formulas.
Lesson Reflections on Congruent and Similar Triangles
How did you feel about this lesson?
I felt that this lesson is one of the harder subjects for this term because there are many forms to prove congruence. And for similarity, the corresponding sides are quite confusing if you relate the wrong sides of the similar triangles. Hence, I feel that this topic is quite hard overall.
What have you learnt in this lesson?
This is also another new lesson to me, so I learned basically every single theorem from scratch. Theories to prove congruence are: SSS (3 sides), SAS (2 sides and 1 included angle), SSA (2 sides and 1 excluded angle), ASA (2 angles and 1 included side) and RHS. Theories to prove similarity are only AA. Even though this may seem confusing, I feel that these are very good theories.
What are your thoughts about this lesson?
I feel that overall this lesson is quite hard and even though the theories may be quite simple, the questions that can come out from these few theories are very difficult and require alot of practice.
I felt that this lesson is one of the harder subjects for this term because there are many forms to prove congruence. And for similarity, the corresponding sides are quite confusing if you relate the wrong sides of the similar triangles. Hence, I feel that this topic is quite hard overall.
What have you learnt in this lesson?
This is also another new lesson to me, so I learned basically every single theorem from scratch. Theories to prove congruence are: SSS (3 sides), SAS (2 sides and 1 included angle), SSA (2 sides and 1 excluded angle), ASA (2 angles and 1 included side) and RHS. Theories to prove similarity are only AA. Even though this may seem confusing, I feel that these are very good theories.
What are your thoughts about this lesson?
I feel that overall this lesson is quite hard and even though the theories may be quite simple, the questions that can come out from these few theories are very difficult and require alot of practice.
Lesson Reflections on Circle Properties
How did you feel about this lesson?
I feel that this lesson is fairly easy, because the theorems are quite common sensical. However, it is difficult to be able to spot the theorems if one is not well practiced. Fortunately, I was always good in visual learning and I was able to see most of the theorems in a second. Furthermore, this topic seems very easy as the theories are very easy to apply and remember.
What have you learnt in this lesson?
Circle Properties is a new subject to me, and mostly I learned all the theories from scratch. For example, theories like Alternate Segment theorem, and tangent is perpendicular to radius are theories that I learned for the first time.
What are your thoughts about this lesson?
I feel that this lesson is one of the easier topics I have learnt. The worksheets already explained what I need to know and the assignments are just to help us get practice for our test, which I think is very important.
I feel that this lesson is fairly easy, because the theorems are quite common sensical. However, it is difficult to be able to spot the theorems if one is not well practiced. Fortunately, I was always good in visual learning and I was able to see most of the theorems in a second. Furthermore, this topic seems very easy as the theories are very easy to apply and remember.
What have you learnt in this lesson?
Circle Properties is a new subject to me, and mostly I learned all the theories from scratch. For example, theories like Alternate Segment theorem, and tangent is perpendicular to radius are theories that I learned for the first time.
What are your thoughts about this lesson?
I feel that this lesson is one of the easier topics I have learnt. The worksheets already explained what I need to know and the assignments are just to help us get practice for our test, which I think is very important.
Target setting
Previous terms
2010 Exam EOY Grade: A1
2011 Exam Term 1 Grade: B3
2011 Exam Term 2 Grade: A1
Current term:
Target Grade: A1
Results obtained:
Test 1: 35
Test 2: 32
Reflections for Test 1:
I was quite happy with my results because this test was rather hard and I was the second in class. However, I realised that I had to get another 35 for the second test if I wanted my overall score to be above 80%, which meant that I may get a possibility of exemption for math. After thinking for awhile, I realised that I was very careless in my questions and if I had practiced a little more, I could have gotten question 3 correct, which was the main killer for everybody. I was also quite fortunate, because this test had been moderated, and if not I would have gotten 33 instead of 35.
Reflections for Test 2:
I was very disappointed with my results because this meant that I would not be able to get my exemption for math anymore, or basically meant that I have almost 0% chance of getting exemption for math. Even though I got 80% for this test, I am not very happy because this test was supposed to be easier. I did not see some properties for the triangles, mainly the Alternate Segment Theorem, which costed me 4 marks. If I had gotten that question correct, I would have got 35, making me eligible for exemption. However, taking the test is not so bad after all. It prepares me for the stress faced in "A" levels, hence I feel that this is a good opportunity to shine.
2010 Exam EOY Grade: A1
2011 Exam Term 1 Grade: B3
2011 Exam Term 2 Grade: A1
Current term:
Target Grade: A1
Results obtained:
Test 1: 35
Test 2: 32
Reflections for Test 1:
I was quite happy with my results because this test was rather hard and I was the second in class. However, I realised that I had to get another 35 for the second test if I wanted my overall score to be above 80%, which meant that I may get a possibility of exemption for math. After thinking for awhile, I realised that I was very careless in my questions and if I had practiced a little more, I could have gotten question 3 correct, which was the main killer for everybody. I was also quite fortunate, because this test had been moderated, and if not I would have gotten 33 instead of 35.
Reflections for Test 2:
I was very disappointed with my results because this meant that I would not be able to get my exemption for math anymore, or basically meant that I have almost 0% chance of getting exemption for math. Even though I got 80% for this test, I am not very happy because this test was supposed to be easier. I did not see some properties for the triangles, mainly the Alternate Segment Theorem, which costed me 4 marks. If I had gotten that question correct, I would have got 35, making me eligible for exemption. However, taking the test is not so bad after all. It prepares me for the stress faced in "A" levels, hence I feel that this is a good opportunity to shine.
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