What are the odds?

My middle school class stumbled upon a nice probability problem that I might just use in my pre-calculus exam.

Oregano

You see, four of my students took cuttings of oregano plants from the pre-school herb gardens when we were studying asexual reproduction, but only three of them grew roots.

That gives a 75% success rate.

$P[success] = 0.75$

The next time I do it, however, I want each student to have some success. If I have them take two cuttings that should increase the chances that at least plant will grow. And we can quantify this.

If the probability of success for each cutting is 75% then the probability of failure is 25%.

P[failure] = 0.25

Given a probability of one plant failing of 25%, the probability of both plants failing is the probability of one plant failing and the other plant failing, which, mathematically, is:

P[2 failures] = P[failure] x P[failure]

P[2 failures] = 0.25 x 0.25

P[2 failures] = 0.25²

P[2 failures] = 0.0625

So the probability of at least one plant growing is the opposite of the probability of two failures:

P[at least 1 success of two plants] = 1 – P[2 failures]

P[at least 1 success of two plants] = 1 – 0.0625

P[at least 1 success of two plants] = 0.9375

which is about 94%.

If I had the students plant three cuttings instead, then the probability of at least one success would be:

P[at least 1 success of 3 plants] = 1 – P[3 failures]

P[at least 1 success of 3 plants] = 1 – 0.25³

P[at least 1 success of 3 plants] = 1 – 0.02

P[at least 1 success of 3 plants] = 0.98

So 98%.

This means that if I had a class of 100 students then I would expect only two students to not have any cuttings grow roots.

An Equation

In fact, from the work I’ve done here, I can write an equation linking the number of cuttings (let’s call this n) and the probability of success (P). I did this with my pre-calculus class today:

$P = 1 - 0.25^n$

so if each student tries 4 cuttings, the probability of success is:

P = 1 – 0.25⁴

P = 1 – 0.004

P = 0.996

Which is 96.6%, which would make me pretty confident that at least one will grow roots.

However, what if I knew the probability of success I needed and then wanted to back calculate the number of plants I’d need to achieve that probability, I could rewrite the equation solving for n:

Start with:
$P = 1 - 0.25^n$

isolate the term with n by moving it to the other side of the equation, and switching P as well:
$0.25^n = 1 - P$

take the natural log of both sides (we could take the log to the base 10 if we wanted, or any other base log, but ln should work just as well).

$\ln{(0.25^n)} = \ln{(1 - P)}$

now use the rules of logarithms to bring the exponent down into a multiplication:

$n \ln{(0.25)} = \ln{(1 - P)}$

solving for n gives:

$n = \frac{\ln{(1 - P)}}{\ln{(0.25)}}$

Note that the probability has to be given as a fraction (between 0 and 1), so 90% is P = 0.9. A few of my students made that mistake.

Lavender

roots-3628 — Roots on two of three plants.

At the same time my students were making oregano cuttings, six of them were making cuttings of lavender. Only one of the cuttings grew.

So now I need to find out how many lavender cuttings each student will have to make for me to be 90% sure that at least one of their cuttings will grow roots.

Students can do the math (either by doing the calculations for each step, or by writing and solving the exponential equation you can deduce from the description above), but the graph below shows the results:

probability-cuttings — Probabilities of successfully growing a lavender cutting.

Here is Today: Our Place in Geologic Time

Here is Today is a neat little interactive website that helps put the geologic timescale, and homo sapiens’ place in it, into perspective.

One Spring’s Month

A month in the spring can make a huge difference. Move your mouse over the image (or click the image) to see the difference between April and May on the Fulton School campus.

The full sized images can be seen here.

Note: To embed the image above use:

<iframe style=”overflow:hidden;” src=”http://earthsciweb.org/js/images/spring/spring.html” width=490 height=326 seamless />

Oil on the Water

oil-water-6633s — A sheen of oil on slowly seeping water.

The organic matter that’s been decaying over the winter is being forced to the surface by the spring’s rising groundwater table. In places of near stagnant water the oily products of decay leave a sheen of oils on the water surface.

oil-water-6635s — Oily sheen on the water.

Red-eared Slider

This little guy was rescued just down the road by one of our bicyclists. His under-shell, which is called the plastron, is beautifully decorated.

It’s in the fish tank with the tadpoles at the moment. Red-eared sliders grow to 12-25cm long, and they’re named after the red splotch that’s located just behind their eye.

It seems happy enough in the tank, but we’ll release him to the creek at the end of the semester in a couple weeks.

They’re native to Missouri, but according to the Missouri Department of Conservation’s nice little reference book, Show Me Herps, these have been the targets of illegal collection, and international trade. Ones released in Europe have become invasive species there.

turtle-3619c2 — The red “ears” aren’t really ears, just a patch of red behind the eye.

Embeddable DNA

ravenclaw-dna — Ravenclaw’s four genes on the DNA string annotated. Note that start and stop codons bracket each gene, and there is non-coding (junk) DNA between each gene.

Using English words like “blue eyes” to represent genes in DNA strings with the DNA Writer runs the risk that students start to wonder if actual genes are coded in English.

I’d say it was a small risk, but today I did have that question from a couple of students today.

Fortunately, it was quite easy to disabuse them of the impression: they didn’t actually believe it, but they just had to know for sure.

I did like one of the questions though, “Does that mean that Spanish people have DNA written in Spanish?”

Embedding the Tiles

With that caveat, since I, and a few of my students, like the pretty patterns the DNA Writer produces (see above), I created a way to embed the color sequences into other webpages like this blog.

By default, the embedded image links back to the DNA Writer website, but you can adjust it so that it does not. Instead, the nucleobase tiles will change color when you click on them. The color changing helps keep track of where you are if you’re trying to string the sequence in beads.

For academic purposes, you can also change the message you get when the mouse hovers over the tiles. By default it give the plain English translation, but you can make it say whatever you want, or even have it just show the base sequence.

The Genetics of Blondes

Blond_Boy_Swimming — Photo by Graham Crumb (via Wikimedia Commons).

Hair color tends to come up pretty organically when talking about heredity and genetic traits. Blonde hair in people of European descent is a result of a the interactions of a combination of genes, but the blonde afros of Melanesians appears to be the result of a single mutation of a single gene.

Switching one “letter” of genetic code-replacing a “C” with a “T”-meant the difference between dark hair and blond hair.

— Loury (2012): The Origin of Blond Afros in Melanesia in Science

Basic JavaScript

Basic Webpage template (using Jquery): basic-jquery.zip

One of my students couldn’t get VPython to install and run her computer. She was running Windows 8, and I have not used Windows, much less this version of it to figure out what the problem was. This is one of the challenges with a bring your own device policy. So, instead I gave the lesson on numerical integration using javascript.

To make things easier, I create a barebones template of a webpage build around javascript (using the jquery library to make interactivity easier).

If you open the webpage file (index.html) in your browser you should see nothing but the word “Hello”. The template is blank, but it’s ready so students can start with the javascript programming right away, which a few of my programming elective students have done.

For reference, this file (basic-jquery-numeric-int.zip) uses the template to create a program that does numerical integration. Someone using the webpage can enter the limits (a and b) and the number of trapezoids to use (n), and the program calculates calculates the area under the curve f(x) = -x²/4 + x + 4.

It’s a very bare template and doesn’t have any comments, so it’s not useful unless you’re at least a little familiar with html and javascript and just need a clean place to start.