# Empirical and Theoretical Probability Worksheets

In this section of our site, we explore the use and nature of several different types of probability. Knowing when to apply the use of one form over another is a goal that we will shoot for here. We spend time working on many different games of chance but will apply this many different applicable situations. These worksheets and lessons help students learn how to read and create probability distributions based mostly on observations.

### Aligned Standard: HSS-MD.A.4

- How Many Hands? Step-by-step Lesson- Sorry to pull poker or playing cards in general into my stats section so much. It does fit right in math-wise.
- Guided Lesson - Android phones, the number of possible hands, six card draw.
- Guided Lesson Explanation - You might want to start by explaining what a "hand" in card games is.
- Practice Worksheet - I try to focus, at the end, on the recognizing the difference between estimated and empirical probability.
- Matching Worksheet - Sorry about letter A and C. I'll come back and fix that one.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

The problems are very tricky. Read the problems slowly and outline them as you go.

- Homework 1 - Mark dealt a hand of fifteen cards from a standard deck of playing cards. The number of possible hands is 4,481,381,406,320. The number of possible hands consisting entirely of red cards (heart and diamond) is 7,726,160.
- Homework 2 - Last month, 205 of my 250 customers purchased one or more of my cell phones. Thus, the probability that a randomly selected customer purchased a cell phone is 205/250 = 0.82. Is this estimated probability or empirical probability?
- Homework 3 - Carl rolls three standard dice. The number of possible outcomes is 216. The number of possible outcomes consisting entirely of {four and five} is 8. What is the probability getting only fives on all three dice?

### Practice Worksheets

The middle sheets is just for understanding the validity of the data.

- Practice 1 - What is the probability of obtaining an outcome that consists entirely of the number four?
- Practice 2 - Last month, 389 of my 560 customers purchased one or more of my super magic brooms. Thus, the probability that a randomly selected customer purchased a broom is 389/560 = 0.69. Is this a theoretical or empirical probability?
- Practice 3 - What is the probability of an outcome that does not consist entirely of black cards?

### Math Skill Quizzes

Quiz 1 has you label the data. Quiz 2 has you determine the actual probability.

- Quiz 1 - Jeff made 45 of his shots out of 70 attempted in this week’s games. His coach determines his probability of making a basket as 0.64. 45/70= 0.64. Is this a theoretical or empirical probability?
- Quiz 2 - A casino craps table pays out when you roll an eleven with a pair of dice. The casino pays out double if you roll a seven. What is the probability of rolling a seven with a pair of dice?

### What are Empirical Probabilities?

The empirical probability is the type of measure where we have to calculate the chances of event's occurrence using the historical data that is available to us. The formula to calculate empirical probability is: Empirical Probability = (Number of times occurred) / (total number of times experiment performed). The probability that we can calculate through this is estimated and are not exact which we calculate using the clearly defined sample space. This is often referred to as the experimental probability because mathematically this is what we should be seeing, but it does not always hold true.

The main advantage of using this measure is that it almost always free of bias and makes no assumptions whatsoever. You will find this this can be helpful when we apply it populations and the decisions that we make and apply to them. Doctors often make many medical tough decisions based solely on patient success rate which based on a very similar process.

### What is Theoretical Probability?

We do not conduct an experiment with this form, we just work the math. Theoretical probability assumes that every single outcome has an equal chance of happening. It directly determined by the sample space that is provided. It is determined by finding the quotient of the number of favorable outcomes divided by the number of possible outcomes: Theoretical Probability = # of Desired Outcomes / Total # of possible outcomes. You will find these values (empirical and theoretical) to be close normally, but as the number of trials that you perform is amplified, they approach equivalency.