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## Balls and Bins

One of my pervious posts mentioned the problem of the balls and the bins. I got this problem from a source on twitter that I have since forgotten. Regardless of its origin, the question has been a fun one to pose to students and colleagues alike (I even asked my in-laws with some very interesting results). For those of you who haven’t read “Practice What You Preach“, The problem is as follows:

You have 8 balls, and 2 bins. 4 of the balls are Red, and 4 of the balls are White. Your job is to arrange the balls in the two bins however you like, but every ball must be put in one of the bins. (ie. no throwing balls away). I will then choose one of the 2 bins, and then draw a ball from that bin. If I draw a White ball, I win; if I draw a Red ball, you win. Which arrangement gives you the best chance at winning the game?
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## Practice What You Preach

I have already expressed my views on the value of probability within the school curriculum. When posed in a creative context, the nature of the subject leads to excellent exploration. I tell this to every class that I teach probability to, and this year my explanation caught up with me.

I gave my first day lesson on a counter-intuitive problem, and then began using experimental probability to verify our results. I reserved the end of class to remind students what we were doing. I got onto my soap box and began preaching the importance of reality; although we were calculating odds, probability is a risk assessment. Lady luck can be fickle.
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## Shouldn’t Probability be Vague?

I have always been drawn to probability because of its mysterious qualities. Maybe it is the result of the online poker fad that swept through my high school during the NHL lockout, but the calculation of odds still grasps my attention to this day. What fascinates me the most is how simple rules such as “AND” and “OR” can quickly create a mess of a situation. What begins in high school (or earlier) as a simple fraction that predicts the toss of a coin, soon balloons into factorials, combinations, Pascal’s Triangle, and Probability Density Functions. Despite the complexity of such calculations, they are still only theoretical; anything could still happen. This is a point that I stress to my students whenever we embark on a study of a game of chance.

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## Merit to Mathematics Labs

There is widespread turmoil among teachers and students when it comes to the practicality of mathematics. School mathematics, at the middle and high school levels, has moved out of the elementary niche of rudimentary skills, but has yet to make it into the realm of complexity necessary to apply it back into the world. Our happy compromise, as teachers, is to go with a two-pronged attack:

1. Tell the students that the practicality comes later
2. Create word problems about trains leaving stations or people tossing balls off cliffs