Chanukah: Experimental vs. Theoretical Probability Dreidl Game

Everyone says "probability" when they hear you played dreidl in math class. I was surprised to wake up Wednesday morning and find I did not have an "Intro to Probability" with dreidls worksheet in my archive. So I made one.

I decided to dig into experimental probability. The main part of the lesson involves the students playing dreidl and recording the results of all spins for their group. I deemphasized the "winning" aspect of it by having them play with poker chips.

Having three groups of students provided wonderful variability in the data. It was fascinating to see how one group had a low yield of gimmels while another had a high yield. This gets into the heart of randomness and variability, and I ended up taking more time than I expected to discuss it.

I reinforced equivalent fractions and estimation by asking students to identify which results were close to 25% (which I arbitrarily defined as 20% to 30%) and which were not. We had two groups with results mostly not near 25% for each possible outcome and one group with really well distributed rolls. 

In class, I only had time to add up the totals for the gimmel column, and we ended up with a grand total of 19 gimmels in 79 class rolls, about 24% and very close to 1/4. 

If I add up the totals for the others, I get: 

(5+4+8)/79 for nun, which is 17/79 (a bit less than 22%); 

27/79 for hey which is a bit more than a third, 

and 16/79 which is about 20%.

So even though in small groups we had varied data, when we added them up the data came closer to the expected 1/4. 

We barely had time on day 1 to talk about multiple events. The next day would be to cover how to find multiple events using a tree diagram and a table (pictured below).

The answer key makes clear that my handout needs more space for a tree diagram. I like tree diagrams because they allow us to compute the probability of 3 or more events. (A table limits you to two.) That way we can ask awesome questions like "What is the probability of getting AT LEAST one gimmel on 3 spins?" 

Teaching them about 'at least' versus 'exactly' is a great extension for strong students.

The tree and table take us to the fundamental counting principle, which is what allows us to multiply the probabilities. Next we would do dependent events, but I don't have dreidl examples for those. 

This lesson is really a rough draft. It went really well; the hardest part was getting the kids to read the directions! Tips, feedback and suggestions always welcome.

Why are there 45 Chanukah candles in a box?

L'ilui nishmat Yechiel Ezra ben Ari Yona, who was talented at mathematics (among many other things.) HY"D.

Most years, when I ask my students this question, I immediately read them the book Hippos Go Berserk by Sandra Boynton. It begins with one hippo, all alone, who calls two hippos on the phone. Hippos arrive in groups of 3, 4, 5, 6, 7, 8, and 9, and charming hijinks ensue. The books ends with the departure of all but the first hippo, and the concluding line, "One hippo, alone once more, misses the other 44."

Usually, at that moment, there is a chorus of "oh!" in the room. Many students have figured out that 45 is the sum of 1 + 2 + 3 .... + 9. On Chanukah, we use 2 candles on the first day (including the Shamash), 3 on the second day and onward until the 8th day where we use 9 candles.

There is a simple formula that allows us to figure this out. Imagine if we group all the days of Chanukah into pairs, with the first and last going together, the second and second last going together, etc. This pattern is sometimes called Atbash in Hebrew, particularly as it applies to letters.

If you "pair off" the days like this, you'll wind up with 4 pairs of sums that all sum to 11. Multiplication tells us that the sum is 44. (I assume the 45th candle is to keep you from calling the company and complaining when one breaks.)

With the hippo example, it's a little trickier. We have an odd number of numbers we're adding. The 1 hippo and the 9 hippos add together to make 10. The two hippos and 8 hippos also add together to make 10. We have 4 pairs of 10 - but the poor 5 hippos (who came overdressed, if you must know) are left without a buddy. That's 40 + 5, or we could think of it as four and a half groups of 10. Either way, there are 45 hippos.

If you'd like a formula for this, you can easily find a formal one online. I just tell the kids sum=(first+last)*the number of pairs, and make sure they understand that the number of pairs is the number of numbers, divided by 2.

For practice, I give silly questions, like:

Mrs. B ate one jelly bean per day of the Omer each day of the Omer.
How many jelly beans did she eat the entire Omer? Answers appear in white after the questions.
(1+49)*24.5=1225
To make it harder, I give them a sequence where the first number is not 1.

Mrs. B did not eat any jelly beans on Pesach. To make up for lost time, she eats 10 jelly beans on day 8 and increases her consumption by one jelly bean per day until the end of the Omer.
a) How many jelly beans did Mrs. B eat on the 49th day of the Omer? (52)
b) How many jelly beans does Mrs. B eat total? (10+52)*21=1302

There's a famous story of a precocious Gauss discovering this formula at the tender young age of 8. With that in mind, I enjoy teaching this lesson to my middle schoolers.

The formula I mentioned dates back to the Pythagoreans. In Jewish tradition, there is a very old use for this in the mishna - in the laws of Menachot.

If you vow to give a mincha of a specific size, but forget what size you promised to bring, what should you do?

A mincha can be brought in integer amounts of issaron, from 1 to 60. You can bring a mincha that is 59 issaron, but not a mincha that is 58.5 issaron nor one that is 61 issaron.

So what should our forgetful pilgrim do? The main opinion in the mishna says we resolve this dilemma by having him bring a mincha of 60 issaron - surely it is as large as, or bigger than, the one he vowed and then forgot. Rebbi (Rav Yehuda Hanasi) says no, he must bring one of each from 1 issaron all the way through to 60 issaron.

How much flour is that? (1+60)*30=1830 issaron.

The Bartenura explains that Rebbi's reasoning is that a 60 issaron mincha cannot fulfill a 1 issaron vow, and the person who made this vow must bring every single offering to ensure he fulfilled his vow. The Bartenura (named after the place in Italy where he lived) did this calculation for us, in the 15th century in his commentary on the Mishna.

Moral of the story? If you must vow, pay attention to what you promised!

Updated to correct errors on 11/24/15.