# High School: Statistics and Probability

### Using Probability to Make Decisions HSS-MD.A.1

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) = nCapaqna, 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.

Here is our recap video on factorials.

#### Drills

1. You've become a mathmaker (a mathematician-matchmaker). A person has walked into your office and asked for their chances of landing a date if they ask 10 random people out and the only responses are either yes or no. Does the probability distribution graph follow a normal distribution if the answers are indeed random?

Yes

This is a time consuming answer, however, it can be quickly solved by recalling the formula P(X = a) = nCapaqna. We know there are only two possible outcomes: yes or no. That means p and q both equal 0.5, yes (or no)? We'll assume these ten random people are nice and good-looking, so a "Yes!" answer is a success, or a. We can calculate for the P(X = a) for each as follows.

 n a p q n – a P(X) 10 0 0.5 0.5 10 0.000977 10 1 0.5 0.5 9 0.009766 10 2 0.5 0.5 8 0.043945 10 3 0.5 0.5 7 0.117188 10 4 0.5 0.5 6 0.205078 10 5 0.5 0.5 5 0.246094 10 6 0.5 0.5 4 0.205078 10 7 0.5 0.5 3 0.117188 10 8 0.5 0.5 2 0.043945 10 9 0.5 0.5 1 0.009766 10 10 0.5 0.5 0 0.000977

Does the probability distribution (probability of X when X = a) follow a normal curve? Yes!

2. What is the sample space for rolling a number on a fair die that that has only four sides? (These dice exist, by the way. They're just tetrahedral.)

{1,2,3,4}

Our sample space is the set of possible outcomes in our experiment. Since our die is four-sided, we can only have four possible outcomes. That means the right answer is (C).

3. You're handed a deck of cards and instructed to randomly pull out all the hearts. You have only thirteen attempts, so choose wisely. What is the random variable in this situation?

The total number of hearts drawn

The random variable is the outcome devoid of bias and due to chance alone. In our case, our random variable is the number of hearts that we pull out of the deck. Our sample space has been defined as {0, 1, 2, 3… 13} hearts drawn from 13 possible attempts.

4. Pick out a coin from your pocket. Preferably a quarter if you have one, as they're easiest to deal with. Go ahead and toss the coin twice. What is the sample space in this experiment?

{HH, HT, TH, TT}

This answer shows all the possibilities after tossing the coin twice. While you might think our options are represented by (B), the order of the coin results matters. All other answers are either incomplete or, as per (D), just show probability.

5. A person stands in one place and shoots a gun at a stationary target without aiming. Assume that his "aim" is random. What is our random variable in this experiment?

Both (B) and (C)

Because the question wasn't very clearly defined, the correct answer is (D). Since the bullet hits a random location, we must measure its relative randomness to something known. In this case, (A) is stationary, so we know where they are all of the time.

6. If your class has 25 students and you wish to randomly sample only those born in August to find out their IQ, your random variable during selection is:

A student born in August

You wish to identify only the ones born in August and then randomly sample from them, so your random variable in this case is some randomly chosen student born in August. You will find out the IQ later.

7. The CEO of a company plans on figuring out how to best assign parking spaces based on the revenues each salesperson generates on his team. He is going to randomly assign a number to all employees and then randomly choose some of them to figure out what the average earnings should be. From there he's going to determine a rank based on all the salespeople's data and assign parking spots accordingly. What is the sample space?

The salespeople in the organization

We can think of all the salespeople as a giant die (which also provides for some strange images). The die has as many sides as salespeople, and since all of our outcomes equate to the sample space, all the salespeople are the sample space. A randomly chosen salesperson equates to one random roll of the giant die, and we're only talking about salespeople, not all employees.

8. A champion archer has been given a mission. He can win \$1,000,000 if he hits the bull's eye 7 times in a row. The calculated probability of any archer hitting the bull's eye is 0.03; however, this man is so good, that his probabilities have increased to 0.04. What is the probability he will hit the bull's eye exactly 7 times in a row?

1.64 × 10-10

If we use the formula P(X = a) = nCapaqna, where n = 7, a = 7, p = 0.04, and q = 0.96, we end up with P(X) = 1.64 × 10-10. This makes sense, as it's difficult enough to hit the bull's eye once, let alone seven times in a row.

9. The archer mentioned in the previous question is now almost certain to not hit his mark 7 times in a row. What are the chances he will hit it 1 or more times in 7 tries? What are the chances he will hit it on his first try?

0.25 and 0.04

You must understand what the question is asking you and what the probability distribution actually shows. The distribution does not imply order. That is, it represents the total number of successes out of a total number of tries, not the order of success. That means that in order to get the total number of successes (in our case 1 or more hits in 7 tries) we must sum all the P(X)'s for a = 1 through a = 7. The sum is 0.25 (approximately). The second part of the question asks us what are the chances he will hit it on his first try. Since one try is an independent event, and we know that means his chance of hitting it on the first try is the same as for any other try, the right answer is 0.04.

10. A random number generator can generate any number between 0 and 100. How many numbers can our generator provide for our sample space?