# Theoretical vs Experimental Probability

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## Objective

SWBAT explore the similarities and differences between theoretical and experimental probabilities.

#### Big Idea

Students will be able to see the relationship between theoretical probability and experimental probability by computing both at the same time.

## Do Now

15 minutes

I’m going to use Holt’s alternate opener for our do now.  This opener gets students looking at different experiments and deciding on whether an event is likely or not to occur.  Since we will be using spinners and dice today, I thought it would be a good idea to have them looking at those items and using the language of probability to describe them.  The students will also need to explain their thinking to a partner once they have completed this worksheet. (SMP 3)

## Vocabulary

20 minutes

Vocabulary

To start the lesson, I’m going to go over some common vocabulary of probability.  I chose 2 words that they may hear throughout this lesson.  Once we have the vocabulary down, I want them to look at Theoretical Probability.  I’m going to tell them what it means, but I want them to think about what it will look like.  So, I will ask them.  By looking at how theoretical probability is written(words), tell me how that will look on paper (numbers).  I’m hoping to hear them tell me it will be a fraction or a ratio (SMP 2)I will also ask them if they recognize any base word in Theoretical?  We have all heard the word “theory” before, what does that mean?  After students have responded to this, I will show them the slide on Experimental Probability.  Again, I will ask them if they recognize a base word? (experiment) and then ask them what does it mean to do an experiment?  I will show them how to write experimental probability (words) and ask them what it will look like in numbers?

Questioning of vocabulary

Before moving on, I will have students do a think-pair-share using the following question:

How are theoretical probabilities alike and how are they different?

Looking for responses like

 Alike Fractions Ratios Dice Spinners cards Different One is what should happen, and the other is what does happen One you act out

After the think-pair-share, allow time for students to share as whole group.

## Theoretical vs Experimental Probability

20 minutes

Next, I’ve constructed the power point so that each slide has an event that starts out by finding the theoretical probability and then finding the experimental probability.  I have the students doing the experiment 50 times, but I may start off by asking them to roll it once and record it.  Make their comparisons. Students may have a difficult time making a comparison especially since the probabilities will be written as fractions.  To help with this, I would encourage students to write their probabilities as decimals and percents too.  Decimals and percents make for much easier comparisons.  Then have them roll it 10 times and make a comparison.  Then roll it 20 times and make a comparison.  I would do this just a couple of times to get students to realize that the more times the experiment is done, the better our chances of getting our experimental and theoretical probability to match.  (when doing the experiment, remind students that they will each get a turn doing the experiment.  Dice, spinners and coins should be passed around the table so all students are engaged in the activity and all students should be recording the results)

## Finding the Complement

15 minutes

After the students have completed their comparisons, I want them to look at the complement of an event.  They have already been given the definition, so I’m going to try and get them to apply the definition of each scenario given.   So, I will say… Knowing the definition of complement,  Make a statement using the complement of “there is a 10% chance of rain today”.  I’m looking for kids to say that there is a 90% chance it won’t rain today so it is likely it will not rain.

If there is a 45% chance of you passing this test, make a statement using the complement.  I’m looking for students to say, “there is a 55% chance that I will not pass the test so it is as likely as not that I will pass”.

Each of these scenarios can be completed independently at first, then shared with partners so the students can hear different ways of expressing the probabilities.

## Closure

15 minutes

Allow students time to write down their thoughts on the following 3 questions.  Reflecting in writing is a good way for students to solidify their learning.  As a whole group, have the students share their thoughts.

Explain whether you and a friend will get the same experimental probability for an event if you perform the same experiment. (I’m expecting to hear that you will not get the same probability because the experiment is based on chance.  It is possible, but unlikely.)

Explain whether you and friend will get the same theoretical probability for an event if you are using the same experiment (this is certain because this is what should happen and they are looking at the same event)

Tell why it is important to repeat an experiment many times. (the more times the experiment is performed the better your chances of matching the theoretical probability)

Common Misconceptions:

Often times, students will write the theoretical probability based on the numerical value.  For example, when asked to find the P(5) on a die.  Students will say 5/6.  When I see this happening, I simply say “ how many 5’s are on the die”, they respond “1”.  I explain that this is the chance of getting a 5, 1/6.