Common Core Standards: Math
1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
We've finally gotten to the coolest part of statistics. You know we're right, but your students might not. Time to convince them.
Students should be able to predict the outcome of an event and express it numerically. That means that if you have ten chocolates of different flavors and you pick one at random, there will be a 1 in 10 chance or a probability of 0.1 that you'll get the one chocolate you actually like. (So it's not Murphy's Law. It's just statistics.)
But life isn't like a box of chocolates (or is it?), and numbers aren't always that pretty. In that case, students should know to use the equation P(X = a) = nCa • pa • qn – a, where P is our probability, n is our total number of trials, X is our event, a is the number of successes of our event, and q is the probability of failure. Also, remember that
(As a side note, those exclamation points are factorials, not our attempt at making things more exciting.)
So why is this the coolest part of statistics? The pretty pictures, of course. Using "graphical displays" just means making charts that represent the probability of our outcome occurring a out of n times, where a goes from 0 to n. That's called a probability distribution, and we usually use histograms (cousin of the bar graph) for these purposes.
For instance, let's take a fair six-sided die. If we roll it 12 times, what is the probability that we'll never get the number 1? Or that all 12 rolls will result in a 1? The probability distribution will tell us all that and then some.
The best way to teach this concept to your students is through as many examples as possible. First, try to see if they can logically understand where the peak of P(X) should occur, and then have them actually calculate and graph it out. With enough practice, they'll be ready to tackle any box of chocolates they come across.