Finite Math—Semester B
Normal is overrated, but normal distributions aren't.
When it comes down to it, life is all about luck. Whether you find twenty bucks sitting on the sidewalk or get struck by lighting, chance plays a bigger role in life than people admit. Even the very best Dungeon Masters and Texas Hold 'Em champions can't escape probability at work. Unless they're cheating.
In this Common Core-aligned course, we'll delve deep into the mysteries of counting theory, normal distributions, why so many people you know share the same birthday, and more. We'd tell you what else we'll cover, but we don't want to ruin the mystery.
On second thought, let's ruin the mystery. With loads of drills, examples, and projects, we'll cover
- counting theory, Venn diagrams, and other ways to deal with large sets of numbers.
- probability, experiments, and the famous Birthday Problem.
- conditional probability, events, and using trees to represent them.
- statistics, histograms, odds, and binomial events.
- normal distributions, z-scores, and game theory.
What are the odds you'll finish this course knowing all there is to know about statistics and probability? We're betting pretty good.
P.S. Finite Math is a two-semester course. You're looking at Semester B, but you can check out Semester A here.
Unit 8. Counting Theory
You learned how to count back in preschool, so it's high-time you stepped up your counting game. We'll start out with sets, unions, and intersections, and even draw a few Venn diagrams. Then, we'll get factorials involved and talk about combinations and permutations and their many applications. By the time we're through, you'll be able to give the Count a run for his money.
Unit 9. Probability
If probabilistic thinking is already intuitive to you, then great. You're ahead of the game. But for the rest of us, this unit is all about making probability as intuitive as how to eat the last Oreo. (Twist-and-lick, obviously.) From setting up experiments to probability properties and formulas, this unit contains more coin tosses and die rolls than you thought were possible.
Unit 10. Conditional Probability
In this unit, we're going to learn how to tell if that third flip being heads depends on the first two being heads (spoiler alert: no). With the help of the Product Rule, trees, and Bayes' Theorem, we'll learn about a new kind of probability: one where the likelihood of an outcome can change in an instant—or at least in the blink of another event happening first.
Unit 11. Statistics
We'll kick off the unit by discussing how to turn any kind of data into a table or graph that you trust. (Those histograms always looked a little shady to us.) After talking about probability and odds, we'll boil down our data into bite-sized summaries, like expected value, variance, and standard deviation. They're not as tasty as those mini-quiches and shrimp cocktails, but we promise they'll be more useful when it comes to data analysis.
Unit 12. Normal Distributions and Game Theory
We won't lie to you: this unit's a biggie. Often, normal distributions can seem more abnormal than you'd think. Don't worry. We'll help you tackle with z-scores, bell curves, and binomial probabilities like a boss. After that, we'll take on game theory, talk about its different strategies, and learn how to kick your opponent's butt at Monopoly. Family game night will never be the same again.
Sample Lesson - Introduction
Lesson 5: Probability Applied
Besides potentially encouraging gambling addictions in classrooms across the nation through the copious use of dice examples, probability can be applied to actual, real-life scenarios as well. Namely, to help us make good choices in life. Such as deciding to skip the craps table after losing five nights in a row. Or helping us to interpret the results of studies published in the news. Or deciding whether or not our secret admirers actually admire us.
Wait, you didn't put that note in our locker?
By combining what we know about biased versus unbiased experimental designs and experimental versus non-experimental probabilities, we are now the new kids on the block. By which we mean: no one's going to put anything by us.
Sample Lesson - Reading
Reading 9.5: Probability Applied
Applying our knowledge of probability can help us to make policy decisions, interpret study results, and keep the school safe from zombies. But really.
Say that about 1 in 1000 high school students are also zombies. Even though this is low zombie incidence rate, kids are obviously spooked. To assuage fears of a zombie apocalypse, the new class president decides to test everyone in the school, the total population of which is about 2000 students, to see if he or she has gone to the other side. The test is 99.9% percent accurate.
Sounds like a good idea, right?
If the test is only 99.9% accurate, then there's some room for error. What about false positives, innocent students who are misidentified by the test as zombies? Let's run a quick probabilistic experiment to estimate how many false positives we might expect.
Another way of saying this is to ask: of students identified as zombies, how many are actually zombies?
First, we can treat the incidence rate of zombies that we're given (1 in 1000) as an experimental probability that someone, say a zombie expert, has provided. Then, in a population of 2000 students, we can expect to find the following number of zombies:
1⁄1000 × 2000 × 99.9% accuracy = 1.998 zombies
The complement of this calculation would be the number of students correctly identified:
999/1000 × 2,000 × 99.9% = 1,996.002 (Note how 1.998 + 1,996.002 = 1,998)
Because correct test results and incorrect test results are complementary events, we know that a 99.9% probability of correctness implies a 0.1% error rate. So to find the number of incorrectly identified students, we have:
999⁄1000 × 2000 × 0.1% accuracy = 1.998 humans.
And for the zombies missed, we have:
1/1000 × 2000 × 0.1% = 0.002 zombies. (See they really complement each other.)
1996 students are correctly identified as humans, 2 are correctly identified as zombies, and two are incorrectly identified as zombies.
Maybe we shouldn't have voted for this class president after all.
So far, we've calculated probabilities as ratios of ways for events to occur. When we apply these probabilities to real-world scenarios, such as a potential zombie apocalypse, we can often think of probabilities as rates or chances.
This can be especially important for interpreting the results of medical tests, or developing real-world strategies. The strategy-making lesson in this case is: pick the strategy that leaves as few innocent students locked up with zombies as possible.
Sample Lesson - Activity
Activity 9.5c: Problem Set
- Credit Recovery Enabled
- Course Length: 18 weeks
- Grade Levels: 10, 11, 12
- Course Type: Basic
Algebra II—Semester A (2014-2015)
Algebra II—Semester B
Finite Math—Semester A
Algebra I—Semester A
Algebra I—Semester B
Just what the heck is a Shmoop Online Course?
Common Core Standards
The following standards are covered in this course:CCSS.Math.Content.1.MD.A.1