Today's Box of Chocolates

by Isabella Weiss

The lesson for today discussed the Two Counting Principles which I think is similar to Choosing which Movements of a Piece to Play at a Recital.

because...

1. Say we have a solo concerto that has four (4) movements: Allegro (1), Adagio (2), Moderato (3), and Presto (4).  Each movement is about four to five minutes long, but due to the time restraints of the recital, you only have enough time for three movements.  In standard performances, you always play a fast movement or a slow movement.  In our case, we have two of each possibilities to be played one after the other: the fast movements (Allegro and Presto), and the slow movements (Adagio and Moderato).  To figure out how many different options you have, you could either write out a list of the different possibilities by hand, or you could multiply the number of fast movements (2) by the number of slow movements (also 2) to find the number of possible combinations.  This would leave you with the answer that there are 4 possible cominations: the Allegro & Adagio, Allegro & Moderato, Presto & Adagio, or Presto & Moderato. This is exactly like the Counting Principle Equation n(A and B) = n(A) * n(B l A). 

 not quite this kind of Presto but...

2.  However, if the recital is running behind schedule and they need to have each performer cut back on their repertoire, you would only be allowed to play one movement.  In this case, you could choose from either category of Fast or Slow movements.  To find out how many options you have (yes, I know it is rather obvious, but we will continue with this example nonetheless), you would add the options from both categories together.  There are 2 Fast movements and 2 Slow movements.  2 + 2 = 4 movements to choose from.  This is similar to the Counting Principle Equation for mutually exclusive events where 

n(A or B) = n(A) + n(B)

3. After much debate about which movement to choose to perform, you still can't make up your mind about which two movements to choose.  Finally, you decide to write down all of the possible combinations, disregarding the "rule" about having one fast and one slow, put them in a hat, and randomly pull out a combination.  Mom, sitting on the side, wants to know how many different ways you could choose a combo of a Fast movement and a Slow movement.  In this case you would simply add the two categories together.  However since there are combinations that have two Fast movements and two Slow movements, you would have to subtract these overlapping combinations from the total.  This will leave you with the answer: there are 4 possible cominations that have a Fast movement and a Slow movement.  This is like the Non-Mutually Exclusive Property: n(A or B)=n(A)+n(B)-n(A∩B).

However,

My comparison is not perfect, because

In reality, you would not be in a situation where you would pull names out of a hat.  Nor would you be likely to play a combination of fast-fast or slow-slow. 

A website that can help:

http://www.mathwarehouse.com/probability/multiplication-counting-principle.php

it has some yummy-looking pictures! 

My personalization: