With the recent emphasis on the study of probability at many secondary school grade levels -- where not so many years ago the topic was relegated to the end of the Advanced Algebra course -- there are many misconceptions that need to be addressed, as well as enlightenments that can, and ought to be introduced. Take for example, the person flipping a coin nine times gets all heads. The usual thinking is that on the next try -- the tenth -- a tail will surely come up. Not true! Each flip of the coin is independent of the previous ones. This is a misconception that ought to be emphasized at the earliest stages of the study of probability.
Then there are many skillful ways to investigate probability questions. Here is a lovely little example that will show how some clever reasoning, along with algebraic knowledge of the most elementary kind, will help solve a seemingly impossibly difficult problem.
Have your students consider the following problem:
You are seated at a table in a dark room. On the table there are 12 pennies, 5 of which are heads up and 7 are tails up. (You know where the coins are, so you can move or flip any coin, but because it is dark you will not know if the coin you are touching was originally heads up or tails up.) You are to separate the coins into two piles (possibly flipping some of them) so that when the lights are turned on there will be an equal number of heads in each pile.
Their first reaction is likely to be: "You must be kidding! How can anyone do this task without seeing which coins are heads or tails up?"This is where a most clever (yet incredibly simple) use of algebra will be the key to the solution.
Let's cut to the quick. You might actually want to have your students try it with 12 coins. Here is what you have them do. Separate the coins into two piles, of 5 and 7 coins, respectively. Then flip over the coins in the smaller pile. Now both piles will have the same number of heads! That' all! They will think this is magic. How did this happen? Well, this is where algebra helps understand what was actually done.
Let's say that when they separate the coins in the dark room, h heads will end up in the 7-coin pile. Then the other pile, the 5-coin pile, will have 5-h heads and 5-(5-h) = tails. When they flip all the coins in the smaller pile, the 5-h heads become tails and the h tails become heads. Now each pile contains h heads! What an awed reaction you will get!
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