Investigating Probability Objective: When done with this lesson, you will have demonstrated how to find and calculate experimental probability for a simple experiment find theoretical probability in a simple situation calculate the probability of the complement of an event when the probability of the event is known Washington State GLE: 1.4.2 Approximate completion time: 2 hours

As many of you know, probability is a specific type of a ratio. Before we venture on to more ratios and the big idea this unit of "proportional reasoning", let's visit this topic of probability since many of you have a good understanding of it already.

Just in case you have never heard before, here is the basic idea of probability:

The "probability of something happening" is

 

That means the probability of something happening could range from 0 to 1 in decimal form or 0% to 100% in percent form. Here's a simple graphic to help illustrate that idea.

Here's a simple example to help refresh our memories about probability.

Suppose that a regular 6-sided die is rolled. What is the probability that: it will be a three? it will be odd? it will be a seven? it will be a number greater than zero?

Answers:

There are six total options and only one is a three. So the probability of rolling a three is 1/6 = .1667 = 16.67% (rounding). There are three odd number on the die. So the probability of rolling an odd number is 1/2 = .5 = 50%. There are no sevens on a single 6-sided die. So the probability of rolling a seven is 0/6 = 0 = 0%. All six of the numbers possible are greater than zero. So the probability of rolling a number greater than zero is 6/6 = 1 = 100%

Does that mean you should expect to roll a three once every 6 rolls? No. Those probabilities listed in the last example are the "theoretical" probabilities. That is, if you rolled the die a significant amount of times, the probabilities would show themselves true. Check out this fun little activity from Cyberchase to see the difference between experimental and theoretical probability .

Did you know the theoretical probability of winning the Poweball is about 1 in 146 million? That doesn't mean someone can't win, but that it's very, very unlikely. Think of it like this. Mark a smiley face on the bottom of a normal playing card and mixed it in with a huge deck of cards (146 million), then start laying out the cards end to end in a straight line with their faces down. You'd have a line about 8,000 miles long! It's possible to pick the right one, just extremely unlikely. What if you didn't want to make an 8,000 mile long line of cards? You could stack that 146 million card deck up nice and straight—over 1,100 feet tall! Good luck picking the "happy" card.

Thinking about picking the right card really helps me to get a grasp on many probabilities. There are many different ways to help understand probabilities better. One such method is to use a simulation.

Suppose you get married some day and have children. (Yes that is a scary thought!) :) If you had two kids, what is the probability that you would have two boys?

Solution:

Theoretically it would be 1/4. That's because there are only four possibilities and one is two boys. (Boy and then another boy, boy then a girl, girl then a boy, and girl with another girl.) Experimentally this may look quite different. Try a simulation to see. Grab a coin. Let's let heads represent boys and tails for girls. Flip it twice in a row. What did you get? (I just got a heads then tails. So I simulated a boy then girl.) Flip your coin twice more. Record your results in this table. Repeat the "simulation" many times (at least 20). 2 boys not 2 boys     I just did it 20 times and had 2 boys only 3 times. That would be an experimental probability of 3/20 = .15 = 15%. Your results will likely be different. Yet if we combined all our efforts and repeated this simulation a significant number of times (maybe 100 or 1,000 or 1,000,000 times), then we'd see the probability approach the theoretical value of 1/4 = .25 = 25%.

Exploration 1:
Open up a new word document to do the following exploration. Be sure to save your work and keep it handy for your assignment. (Remember to copy and paste in the directions and to do your work in a clean and distinguishable font—perhaps blue.)

According to a survey of students in Massachusetts, "In all, 10 percent of the students surveyed said they smoke cigarettes daily, down from 15 percent in 1995. Daily smoking also increased with grade level, from 6.6 percent among ninth graders to 17 percent among high school seniors".

1. Make a list of at least 20 people that are 10 graders or younger (either from this class or your friends or made up).
2. Find a 6-sided die. (Rolling a 1 on the die has a probability of 1/6 which is about 17 percent.)
3. For each of the student in your list, roll the die. Record what number each student had for his or her roll.
4. Label each of the students who had a one for his or her roll as "smoker". Label the others as "non-smokers".
5. Explain how this simulation helps represent the propability that smoking increases to about "17 percent among high school seniors".
6. Explain what limits you think this simulation has on the reality of smoking for those students on your list.

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Exploration 2:
In your word document to do the following exploration. Be sure to save your work and keep it handy for your assignment.

Roughly 1/3 people in the U.S. die each year from some sort of heart disease. You likely know of someone who has died from heart disease or will know of someone who will die from heart disease. (That is part of the sad reality of life.)

1. Create a list of at least 20 people of differing ages.
2. Describe a method to simulate a "1/3" probability.
3. For each person in your list, simulate whether or not they will die from some sort of heart disease.
4. Label each person in your list as "heart disease death" or "other death".
5. What percentage of your people were simulated to have a heart disease death? Explain why this percentage will not be exactly 33.333% (1/3 of the people) each time the simulation is done.
6. Explain how you think this simulation is valid or not valid in predicting who will die of heart disease.
7. Explain some factors or influences outside the realm of probability that would determine whether or not a person in the U.S. would die from heart disease.

Most of you recognize that although 1/3 of all the people in the U.S. may die from a type of heart disease, there are 2/3 of the people who will not.
That other part of the whole is called the "complement". Many times it is simpler to find the probability of something by first finding the probability of the complement. Take a look at this next example and see when using the complement might be easier.

Bengal Tiger Blue Jay Bumblebee Galaxy Giraffe Jungle
Situation:	Marvin Bull placed a large order for some marbles at Megaglass.com. He ordered 25 Bumblebees, 15 Blue Jays, 10 Bengal Tigers, 15 Giraffes, 5 Galaxies, and 20 Jungles. He ordered all of them in the 16mm size. When the ordered arrived at his house all the marbles were in one bag. He was extremely excited. He stuck his hand into the bag and drew out one marble without looking. What is the probability that: it was a Bumblebee? it was a Giraffe? it was something that started with a "B" or "G"?
Solution:	There were 25 + 15 + 10 + 15 + 5 + 20 = 90 marbles in total. The probability that the one marble drawn at random was a Bumblebee is 25/90 = .2778 = 27.78% (when rounded) since there were 25 Bumblebees in the total 90 marbles. a Giraffe is 15/90 = .1667 = 16.67% (when rounded) since there were 15 Giraffes in the total 90 marbles. something that started with a "B" or "G" could found by adding up all the Bengal Tigers, Blue Jays, Bumblebees, Galaxies, and Giraffes. Yet that is almost all the marbles! Why not simply subtract the number of Jungles from the total? There are 90 total and 20 Jungles. That leaves 70 non-Jungles. So the probability of drawing a non-Jungle (which is something that started with a "B" or "G") is 70/90 = .7778 = 77.78% (when rounded).

 

In summary, the basics of probability that we are looking at are

1. Probability is a ratio.
2. Theoretical probability is a ratio.	The theoretical probability of an event E occurring is
3. Experimental probability is a ratio.	The experimental probability of an event E occurring is
4. Complementary events have probabilities that sum to 1.	P(E c) = 1 − P(E) (This is read "The probability of the complement of E is 1 minus the probability of E".)

 

Now try your hand at some more probabilities by downloading and completing this assignment.