Acharei Mot/Kedoshim- Probability

"[Aaron] shall take the two he-goats and stand them before Hashem, at the entrance of the Tent of Meeting. Aaron shall place lots upon the two he-goats: one lot 'to Hashem' and one lot 'to Azazel.' Aaron shall bring near the he-goat designated by lot to Hashem, and he shall make it a sin-offering. And the he-goat designated by lot to Azazel shall be stood alive before Hashem, to atone upon it, to send it to Azazel to the wilderness." ~Vayikra 16:7-10

Rashi on 16:8 explains what is meant by placing lots on the goats. He explains that one goat was placed to Aaron's right and the other to Aaron's left. He had a lottery box with the two lots (presumably these were labels or markers of some sort)- one lot indicated to Hashem, and one lot indicated to Azazel. Aaron placed both his hands into the box and picked out one lot in each hand. The lot in his right hand labeled the goat on his right, and the lot in his left hand labeled the goat on his left. 

Based on this explanation, we understand that until the lots were chosen by Aaron, either goat could be the one going to Hashem and either goat could be the one going to Azazel. This leads us to the concept of probability.

I wrote about probability once before, in my post last year on Parshat Beshalach. There, I explained probability as follows:

Definition- the chance that something will happen (merriam-webster.com)
Probability is expressed as a fraction between 0 and 1, where 0 means that there is no chance for the event to happen and 1 means that the event will definitely happen. For clarification, chance and probability are the same, however, chance is the number expressed as a percentage and probability is the number expressed as a fraction.

In my previous post, I explained how to calculate compound probabilities. Here, I would like to look at explaining simple probability for younger students. 

We can think about this as simply as possible through some basic examples:

Case #1:
If you have 2 marbles in a bag- one blue and one red- and you want to pick out one marble, you can not say that you will certainly pick the blue or certainly pick the red. You have an equal chance of picking out one or the other. 

How do we show this mathematically?
We can express the probability as a fraction where the denominator tells us how many total options we have, and the numerator tells us how many opportunities we have for picking the item that we want. 

In this case, we have:
*2 marbles (our denominator) and 
*if we prefer the red marble, we have 1 of those in the bag (our numerator)

Therefore, the probability of picking the red is 1/2. 
This could be expressed as 50% chance of picking the red. 

If we wanted to pick the blue marble, there are still the same 2 marbles in the bag and only 1 of the blue in the bag, so the probability of picking the blue is also 1/2. 

Parsha Connection Interjection:
With this understanding, we see that each goat has a 50% chance of being sent to Hashem or to Azazel. They have equal chance of being sent to for either purpose.

More on Simple Probability:
As long as there are the same number of each type of item, then the probability of picking any of the items will be the same as the probability of picking any of the other items. 

Case #2:
If you have 12 marbles in a bag, with 6 red and 6 blue, then the probability of picking 1 marble of either color would be 6/12, or still 1/2 (when the fraction is reduced). 

Case #3:
What if you had 12 marbles in 4 different colors? 
If you had 3 marbles of each color- blue, red, yellow, and green- then the probability of picking one in any of those 4 colors would be 3/12, or 1/4 when reduced. 

Case #4:
What happens if we don't have the same number of each color? 
Let's say we had 10 marbles:
*5 green
*2 red
*2 blue
*1 yellow. 
What would be the probability of picking each color when picking 1 marble?

Green- 5/10 (reduces to 1/2)
Red- 2/10 (reduces to 1/5)
Blue- 2/10 (reduces to 1/5)
Yellow- 1/10

So, we have the best chance of picking a green, a smaller, but equal chance of picking red or blue, and it is the least likely that we'll pick yellow. 

Activities to Test Probability: (Theoretical vs. Experimental Probability)
There are many similar ways of testing probability:
*flipping a coin (chance of heads or tails)
*rolling a die (chance of rolling numbers 1-6, or 6 colors, or 6 other labels)
*picking items unseen from a group (it's important that the objects aren't distinguishable by size, or touch)

Theoretical Probability is what the probability is for picking an item based on calculation.
Experimental Probability is what the probability looks like as you actually test picking item.

Students can chart their probability findings by setting up a bar graph on large graph paper (eg 1in x 1in squares) or large chart graph paper on an easel. If they are tracking the probability of heads vs tails on a coin, they would have a column for each result, and flip the coin multiple times. Each time they flip, they color in 1 square on the chart above the corresponding picture (heads or tails). Different students can test this individually and then compare their results. If you collate all the graphs from the class onto a single graph, students should start to see that the more trials they have, the closer experimental probability comes to looking like theoretical probability.

Thinking about the difference between theoretical and experimental probability- Just because you flip a coin 100 times doesn't mean that you will have exactly 50 heads results and 50 tails results. Some people might have 50-50, while another might have 48-52, and a third person could have 40-60. But if after 4 flips you have 3 heads and 1 tails, you could still end up with 50 heads and 50 tails after 100 flips.