Walnut Canyon

Today I drove out to Walnut Canyon and did some hiking and sightseeing. The Island trail is a really steep 1 mile loop with lots of stairs. It goes down into the canyon and you can see some cliff dwelling rooms where the Indians used to live. The Rim Trail is an easier .7 mile trail that runs along the rim of the canyon. I did both trails in addition to going to the visitor center. Bonus: I had no idea before I decided to go, but today was a fee free day in honor of National Public Lands Day. What are the odds?!?!

Can you pass MAT 114?

This is the exam I gave to my MAT 114 students on Monday. They had 50 minutes.
Exam 1
How'd you do? The average in my class was around 77%. It took about 4.5 hours to write but only 1.5 hours to grade.

A Thought-Provoking Question

What would you call the milk that's been poured on cereal? Is it a broth, sauce, or beverage?

Since I'm the instructor of a course, I have to have office hours so that if my students have questions, there are designated times that I'm in my office and they can come talk to me. But I'm teaching a lower division course and students never come to office hours, so it's basically time that I have to be in my office but no one comes with questions (I promise this is relevant to the above question.)
Today was one such day. Alex and I were sitting in our office during office hours and he brought up this question. So he and I (and Ian for a while) debated it and came to no agreeable conclusion. So we decided to put the blackboard in the office to good use. This is what we came up with:

So in conclusion, the milk that's been poured on cereal can be a beverage and/or a sauce but not a broth. Mystery solved, crisis averted.

Friends :)

So I thought I'd freak you all out by starting with this picture. It's at the Grand Canyon.

Anyway, Natalie and Marissa came to visit for the weekend. It was so much fun having friends!

We did some hiking in Williams, did some shopping in the downtown area, and went to the Grand Canyon.

Yes that's me. Yes they straightened my hair. Yes it took forever!

We randomly saw this elk when we were walking along the rim back to the car. It literally came out of nowhere!

If these clouds look ominous, it's because they were. We got caught in a giant thunderstorm at the Grand Canyon. There was hail and everything. Luckily...

 

we found this ledge to wait out the storm under.

Step 1: Find some random wheel barrows along the trail
Stem 2: Wonder inwardly (or outwardly) why there are wheel barrows along the trail
Step 3: Take funny picture

If you haven't been to the Grand Canyon, you should. It's amazing!

At the top of Bill Williams Trail. In case you didn't know, this is what Northern Arizona typically looks like.

Still at the top of Bill Williams Trail. This one's kind of dark, but the clouds are cool.

Downtown at Criollo Latin Kitchen.

Monty Hall Problem

So someone (I won't mention who to prevent grievous bodily harm to this person) requested that I post about what I'm doing in my classes and such.  So that's what today's post is about. Consider yourself warned.

Picture this: Your the contestant on a game show with host Monty Hall. Monty asks you to choose one of 3 closed doors. Two doors hold goats and one holds $50,000. You choose a door. Monty then opens one of the doors you did not choose. If the opened door contains a goat, Monty asks if you would like to switch doors. What do you do?
Does it make a difference if Monty knowingly opens a door with a goat behind it?
Think about it...

Instead of telling you the answer straight up, I'm going to explain a few things first just to torture you.
To give you a little background, conditional probability of A given B means the probability of event A given that event B has occurred. Also, the conditional probability of A given B equals the probability of A and B divided by the probability of B. Notation-wise, P(A) denotes the probability of event A occurring. Also P(A|B) denotes the conditional probability of A given B. Also also, P(A|B) = P(A and B)/P(A).

So back to Monty Hall.  Let's assume:
W = the event that you win the $50,000
G = the event that a goat is revealed
M = the event that the money is revealed
Since there are 3 doors and only one holds $50,000; the probability of winning is 1/3, or using our handy notation, P(W) = 1/3.

If Monty randomly opens one of the un-chosen doors then either the money is revealed (denoted M) or a goat is revealed (denoted G).
So P(W) = P(W and G) + P(W and M).
Using the above information about conditional probabilities and doing a little algebra:
P(W) = P(W|G)*P(G) + P(W|M)*P(M)
We know that P(W) = 1/3, P(G) = 2/3, P(M) = 1/3 and that P(W|M) = 0 (because you cannot win if Monty opens a door that you did not chose and it reveals the money.) Filling in what we know, we now have
1/3 = P(W|G)*(2/3) + 0*(1/3)
Using simple algebra, we find that P(W|G) = 1/2.
This tells us that if we chose a door and Monty randomly opens one of the un-chosen doors and it reveals a goat, it doesn't matter if we switch doors or not.

Now what if Monty knowingly opens a door with a goat behind it? Should this affect our decision to switch doors? Let's find out.
So we still have P(W) = P(W|G)*P(G) + P(W|M)*P(M) just like in the last part, but now some of the probabilities will be different because Monty knows which doors have goats and will select one of those to open. P(W) = 1/3, P(W|M) = 0, but now P(G) = 1 because Monty always opens a door with a goat in this scenario and for the same reason, P(M) = 0. So we have:
1/3 = P(W|G)*1 + 0*0
Now we see that P(W|G) = 1/3.
This tells us that if we chose a door and Monty knowingly opens one of the un-chosen doors to reveal a goat, we have a 33.3% chance that our door contains the money and a 66.7% chance that the door that we didn't chose holds the money. In this scenario it would be in our best interest to switch doors.

So in on scenario it doesn't matter if we switch or not and in the other, it is in our best interest to switch. Bottom line: switch doors!

Tastier Cookie Experiment

So today after doing some climbing and playing soccer, I went back to something I'm relatively good at: baking. I may or may not have eaten all the cookies I made a few days ago and have been craving some more. I decided to try out another new, healthy recipe. Here's how it went:

Step 0: Buy ingredients.The only ingredients I had were vanilla, almond meal, and salt all of which I bought for the last batch. Sorry no picture.

Step 1: Mix wet & dry ingredients separately

 Step 2: Mix and add chocolate chips

 Step 3: Drop on parchment paper lined baking sheets

 Step 4: Bake!

I would say these cookies are tastier than the ones I made a few days ago. They have coconut, a better texture, and more complex flavor.

Here's the recipe:

Nikki's Healthy Cookies
Indredients:
3 large, ripe bananas, well mashed (about 1.5 cups)
1 t. vanilla
1/4 c. coconut oil, warmed just so it's not solid
2 T. agave nectar
2 c. oats
2/3 c. almond meal
1/3 c. unsweetened shredded coconut
1/2 t. cinnamon
1/2 t. salt
1 t. baking powder
6-7 oz chocolate chips

Directions:
Preheat oven to 350.
In a large bowl mash bananas and add vanilla, coconut oil, & agave nectar. Set aside. In another bowl whisk together oats, almond meal, coconut, cinnamon, salt, & baking powder. Add the dry ingredients to the wet and stir until combined. Stir in chocolate chips. Drop dollops of dough on a parchment paper lined baking sheet.. Bake for 12 - 14 minutes.
Makes 2-3 dozen

Tasty Cookie Experiment

I've been wanting to try out some healthy cookie recipes for a while now. Flagstaff is a pretty hippie-ish town that has more Natural/Healthy markets than regular ones (not really, but it's pretty close).  So today I hit up New Frontiers Natural Market and stocked up on the essentials (almond flour & agave nectar) and started baking.

Step 1: Get out all the ingredients

 Step 2: Mix wet and dry ingredients separately

 Step 3: Mix all ingredients & add chocolate chips

 Step 4: Form cookies

 Step 5: Bake

 Step 6: Eat! Best when served warm with a glass of milk

Overall, I would call this experiment a success. These were some of the easiest cookies I've ever made. No mixer required, which is a good thing because I don't have one! The only catch is that a few of the ingredients aren't normally in the pantry and they are a little expensive/hard to come by.
Bottom line: I will be making these again!

In case you're wondering, here's the recipe:

Almond Chocolate Chip Cookies
Ingredients:
1 c. almond flour/meal (ground up almonds, available in most natural markets)
1/4 t. baking soda
1/8 t. salt
1/8 t. cinnamon
2 T olive oil
2 T agave nectar
1/2 t. vanilla
1/4 c. chocolate chips

Directions:
Mix dry ingredients and wet ingredients in separate bowls. Add wet ingredients to dry ingredients and stir until combined. Add chocolate chips. Drop by spoonfuls onto baking sheet. Bake for 7 minutes at 350 on parchment-paper-lined baking sheets. Makes about a dozen cookies.