Minus' Solution to

Extract the Protractor
Posted January 6 - 12, 1997

The probability is 1/3.

Call the clear protractors Clear1 and Clear2.

Here are the following possibilities:

    	  Amelia         Cory
         Clear1         Clear2
         Clear2         Clear1
         Gold           Clear1
         Gold           Clear2
         Clear1         Gold
         Clear2         Gold

Two of the six possibilities, with each of the six being equally likely, will result in neither party getting the gold protractor. 2/6 = 1/3.

Your Solutions

Mr. T and his little Einsteins, a class of Algebra 1 freshmen and sophomores at Archer City High School in Texas, gained some extra practice with probability. The DEN crew also enjoyed Mr. T's digression. We, too, are trying to calculate the probability of a free day!

My students and I did this problem last night as our "Problem Of the Day" (POD). Although most of my students are Algebra 1 freshmen and sophomores and we haven't spent much time this year on probability (except for them figuring the probability of us "having a free day",which of course has a probability rapidly approaching zero.) But I digress; here is our solution to "Extracting the Protractor".

Probability without replacement - Amelia, for example, will choose her protractor first, giving her a 2/3 probability of not getting the golden one; Cory, on the other hand, only has two protractors left to choose from, giving her a 1/2 probability. Therefore, the probability of neither of them selecting the golden protractor, without replacement, is 2/3 * 1/2 = 1/3.

Mr. T and his little Einsteins also proposed another way of looking at the problem:

Suppose, they said, one person was allowed to choose from the three protractors. The chosen protractor would then be replaced, and the other person would also get a choice from all three protractors.

In this scenario, it would be possible for both Amelia and Cory to get the golden protractor, which they probably deserve anyways. However, it would be more likely that neither got the golden protractor, as there would be a 2/3 chance that Amelia didn't get the protractor, and a 2/3 chance that Cory didn't get the protractor. Multiplying (2/3)(2/3) gives you 4/9, which means that there is a 4/9 chance that neither Cory nor Amelia gets the golden protractor.

Minus liked John Moser's [JohnMoser@msn.com] point regarding the difference in weight between gold and plastic. It's back to the old periodic table for our math shark.

These solutions presume that they know nothing about Chemistry. I think they know that gold is heavier than plastic. Therefore, even with the gloves they could recognize the heavier protractor. This means that they should be able identify the right protractor all the time. In this scenario they succeed 100% of the time and would not find the golden protractor 0% of the time. It's not Mathematical but it is surely a possibility.

And finally, Terry [TerryS2323@aol.com] provided Minus with a short and sweet answer:

The odds are 2/6= 1/3 that they will not pick the golden protractor.

Thanks to this week's creative
feeders, Minus now realizes that
there is more than one way
to approach a math problem.