Ayşe Naz's Second Page

Using Pascal’s Triangle for Binomial theorem expansion:

Fifth Post

Look at these two and can you see the similarity?

       

 Yes the coefficients we use in the expansion (they are called binomial coefficients)  come from the Pascal’s Triangle. As we can see that the first term in the expansion of (a+b)ⁿ is always 1aⁿb⁰ which is nC0anbn0  and the second term is nC1a(n-1)b1. There is a pattern in here the coefficient starts form nC0 and goes until nCn and the exponents add up to n all the time. Exponent of “a” decreases by 1 and the exponent of “b” increases by 1 (please keep in mind that the first term is aⁿb⁰)

It is very easy to expand a binomial when we have a Pascal’s Triangle but we can expand it without the Pascal’s Triangle but using his theorem. Lets expand (a+b)⁶

n=6

6C0a⁶b⁰+6C1a⁵b¹+6C2a⁴b²+6C3a³b³+6C4a²b⁴+6C5a¹b⁵+6C6a⁰b⁶

Now look at the 6^th row of Pascal’s Triangle.

Sometimes watching a video helps you understand a topic better. I’ve found this video and thought it would be helpful

https://www.khanacademy.org/math/algebra2/polynomial_and_rational/binomial_theorem/v/pascals-triangle-binomial-theorem

 

Combination with Ice cream cones:

6th Post

In this activity you have 6 flavors that you can choose from. You have to choose only 2 flavors and it doesn’t matter if the chocolate is in the right or left vice versa. How many choices do you think you have? Now check your answer by playing this game and let’s see how many choices you have.

Play the game at :http://www.transum.org/Software/SW/Starter_of_the_day/starter_November12.ASP

You have exactly 15 choices. Since the arrangement is not important it’s a combination problem. You have 6 flavors in total and you can only choose two so here is the solution:

6C4= 6! / (2!*4!)

A note: We actually never did this online activity in class but we solved problems about ice cream cones but I’ve found this one online and thought it would be helpfulJ

 Monthy Hall Problem:
7th post

There was a very famous Tv show called “Lets make a deal” in 1960s and the Tv show was like this: There are three doors that a person can choose from. The host knows exactly what’s behind the doors. There are two goats and a car behind those doors. You pick a door, lets say you pick door number two and then the host comes and opens the door 3, revealing a goat. Now you are left with door 1 and door 2. The host asks you if you would like to switch the door. What would you do?

Keep your answer in mind and now play this game first without switching and then with switching. Keeping a track of your results may be useful.

http://math.ucsd.edu/~crypto/cgi-bin/MontyKnows/monty2?1+12892

What did you get? I’ve played this game 50 times and kept a record here’s what I’ve got:

There was a pattern in the game and I’ve figured it out in the middle so my answers are not quite accurate but Ben from the movie 21 can explain this question for us:

https://www.youtube.com/watch?v=Zr_xWfThjJ0

Now lets hear the rationale behind this game from a real mathematician:

https://www.youtube.com/watch?v=mhlc7peGlGg

Now take a look at this tree diagram:

I highlighted “Loose” with pink and “Win” with yellow highlighter.

Ayşe Naz’s Method:

8th (Last) post

In my class when we encounter a problem with a complementary event we say “solve it with Ayşe Naz’s Method”. So in my very last post I am going to explain how to solve questions with “Ayşe Naz’s Method”

First lets define the complement of an event firs: It means that the probability of A not occurring.

We can show it mathematically like this:

P(A’)= 1- P(A)

Here is a question with a complementary event.

Q: One card is selected from a deck. What is the probability the card is not a diamond?

As you can see from the picture below every suit have 13 cards. In a deck there are 52 cards and we have 13 diamonds in a fair deck.

Lets first find the probability of getting a diamond:

It’s 13/52 which is ¼.

Now let’s subtract this from all(1) and the final  answer is ¾ .

You might want to watch a video on complementary events so here is a video:

https://www.youtube.com/watch?v=7oDkPOE-Fek

 A note: Always keep in mind that when you see a question with "the probability the card is not a diamond" or "the probability of a student is not a girl" you should use Ayşe Naz's Method because it is easier.