SRB – Montessori Muddle

Planting Probabilities

The Gardening Department of our Student-Run-Business sowed seeds in little coconut husk pellets. The question was: how many seeds should we plant per pellet.

Since we’ll only let one seedling grow per pellet, and cull the rest, the more seeds we plant per pellet, the fewer plants we’ll end up with. On the other hand, the fewer seeds we plant (per pellet) the greater the chance that nothing will grow in a particular pellet, and we’ll be down a few plants as well. So we need to think about the probabilities.

Fortunately, I’d planted a some tomato seeds a couple weeks ago that we could use for a test case. Of the 30 seeds I planted, only 20 sprouted, giving a 2/3 probability that any given seed would grow:

$P[\text{grow}] = \frac{2}{3}$

So if we plant one seed per pellet in 10 pellets then in all probability, only two thirds will grow (that’s about 7 out of 10).

What if instead, we planted two seeds per pellet. What’s the probability that at least one will grow. This turns out to be a somewhat tricky problem–as we will see–so let’s set up a table of all the possible outcomes:

Seed 1	Seed 2
grow	grow
grow	not grow
not grow	grow
not grow	not grow

Now, if the probability of a seed growing is 2/3 then the probability of one not growing is 1/3:

$P[\text{not grow}] = 1 - P[\text{grow}] = 1 - \frac{2}{3} = \frac{1}{3}$

So let’s add this to the table:

Seed 1	Seed 2
grow (2/3)	grow (2/3)
grow (2/3)	not grow (1/3)
not grow (1/3)	grow (2/3)
not grow (1/3)	not grow (1/3)

Now let’s combine the probabilities. Consider the probability of both seeds growing, as in the first row in the table. To calculate the chances of that happening we multiply the probabilities:

$P[(\text{seed 1 grow}) \text{ and } (\text{seed 2 grow})] = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$

Indeed, we use the ∩ symbol to indicate “and”, so we can rewrite the previous statement as:

$P[(\text{seed 1 grows}) \cap (\text{seed 2 grows})] = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$

And we can add a new column to the table giving the probability that each row will occur by multiplying the individual probabilities:

Seed 1	Seed 2	And (∩)
grow (2/3)	grow (2/3)	4/9
grow (2/3)	not grow (1/3)	2/9
not grow (1/3)	grow (2/3)	2/9
not grow (1/3)	not grow (1/3)	1/9

Notice, however, that the two middle outcomes (that one seed grows and the other fails) are identical, so we can say that the probability that only one seed grows will be the probability that the second row happens or that the third row happens:

$P[\text{only one seed grows}] = P[(\text{Row 2}) \text{ or } (\text{Row 3})$

When we “or” probabilities we add them together (and we use the symbol ∪) so:

$P[\text{only one seed grows}] = P[(\text{Row 2}) \cup (\text{Row 3}) \\ = \frac{2}{9} + \frac{2}{9} = \frac{4}{9}$

You’ll also note that the probability that neither seed grows is equal to the probability that seed one does not grow and seed 2 does not grow:

$P[\text{neither seed grows}] = P[(\text{seed 1 does not grow}) \cap (\text{seed 2 does not grow}) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$

So we can summarize our possible outcomes a bit by saying:

Outcome	Probability
both seeds grow	4/9
only one seed grows	4/9
neither seed grows	1/9

What you can see here, is that the probability that at least one seed grows is the probability that both seeds grow plus the probability that only one seed grows, which is 8/9 (we’re using the “or” operation here again).

In fact, you can calculate this probability by simply taking the opposite probability that neither seeds grow:

$P[\text{neither seed grows}] = 1 - P[\text{neither seed grows}]$

Generalizing a bit, we see that for any number of seeds, the probability that none will grow is the multiplication of individual probability that one seed will not grow:

Probability that no seeds will grow

Number of seeds	Probability they wont grow
1	1/3	(1/3)¹
2	(1/3)×(1/3) = 1/9	(1/3)²
3	(1/3)×(1/3)×(1/3) = 1/27	(1/3)³
n	(1/3)×(1/3)×(1/3)×…	(1/3)ⁿ

So to summarize, the probability that at least one plant will grow, if we plant n seeds is:

$P[\text{at least one seed grows}] = 1 - P[\text{no seeds grow}]$

which is:

$P[\text{at least one of n seeds grows}] = 1 - P[\text{1 seed grows}]^n$

Which is something we may have seen before: What are the odds?

Finally to answer our question: how many seeds we should plant, we need to decide how high a probability we need of success:

Probability that at least one seed will grow

Number of seeds	Probability that at least one seed will grow	%
1	2/3	67%
2	8/9	89%
3	26/27	96%
4	80/81	99%
n	1-(1/3)ⁿ

The Head of Gardening leads the planting of seedlings.

CHICKEN MIDDLE’S FIRST EGG!!!

Last year, our middle schoolers named their business Chicken Middle. I was a bit skeptical, but the name stuck. This year, thanks to a lot of help from the school community (thanks to the R’s for the Ruby Coops), we finally have chickens (thanks to Mrs. C. for fostering chicks for us over the summer).

And today, we had our first egg. The students were a little excited.

It looks a little lonely sitting there by itself in the egg carton (thanks to Mrs. D., Mrs. P., and everyone else who donated egg cartons), but with a little luck it will have lots of company soon.

A student hand-feeds crickets to the chickens.

The Eggs have Arrived

After waiting an eternity (about two weeks) the Middle School business’ eggs have arrived.

We set up the incubator downstairs in the pre-school/Kindergarden classroom so Mrs. D’s kids will have the chance of monitoring them. The little kids will be responsible for turning the eggs, while the middle schoolers have set up a data logger and a couple temperature probes to keep track of the temperature in the incubator.

The incubator was provided by Ms. Mertz. It’s put together out of plywood with a 75 W incandescent light bulb as the heat source. Unfortunately there is a significant thermal gradient and although we salvaged a couple of computer fans for the purpose we did not get around to installing them –and more importantly testing them– in the incubator before the eggs arrived.

We’ll see how it goes.

Quote for the Day: On Power

The measure of a man is what he does with power. — attributed to Plato

It’s quite fascinating how character traits are highlighted when students gain the rights and responsibilities of the student run business supervisor. Certainly, some students become a bit over-enthusiastic about exercising their rights; though that’s never been much of a problem for the main supervisor because I try to make sure that anyone who gets to be the main supervisor has spent some time supervising a division. Also, Montessori students get a lot of practice working in their small groups, so leadership positions are usually not too much of a shock to them. Those that do try to throw their weight around excessively, provide the class with the opportunity to discuss worker rights, and a deepening of their understanding of the needs for checks and balances.

What I find most interesting, however, are the students who see only the responsibility of leadership and get bogged down and stressed out trying to manage all the details. For them the practice of leadership does a lot to help build character.

Right hand “man”

Lunch on Wednesdays follows our main block of Student Run Business time. It’s after they’ve delivered pizza, prep-ed for a week of bread, completed finance and its reports, prepared and processed order forms, and sorted out the plants.

Over the last couple weeks I’ve started having my students discuss the business over lunch (including finance reports presentations) and it’s turning into a regular board meeting.

Today they started assigning seating.

We usually sit around two long tables set end to end, with the main supervisor on one end and myself at the other. Today the main supervisor started laying out plates and positions. Pizza supervisor to his right, bread to his right, finances one down from bread and sales across from finances. Everyone else could find their own spot.

I was a little surprised at this unprompted expression of hierarchy. Pizza is our most involved part of the business and the core of the the enterprise so its supervisor, P., has a very important post. She was placed on the right hand of the main supervisor!

I asked the main supervisor why he did it. He said, “I don’t know.” I even had to explain the meaning of the term, ‘right hand “man”‘.

It ended up with the supervisors at one table and everyone else (and myself) at the other.

Except for the plants supervisor. Plants have been going slowly, lately, including some seedling failures. The plant supervisor sat all the way down the table, next to me.

…

I can feel it in my bones that there are some interesting lessons in all this. From organizational structure to non-verbal communication.

But since we’re dealing with positions around a table, and we’ve been talking about the importance of place in geography, the best context to discuss this right now might just be one of the importance of geography and place in the interactions among people.