4.1: Empirical Probability
- Page ID
- 58262
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Define empirical probability as based on observed outcomes from experiments rather than theoretical calculations.
- Calculate empirical probability using the ratio of favorable outcomes to total trials.
- Understand the law of large numbers and its role in stabilizing probabilities with increased trials.
- Define key terms such as sample space (all possible outcomes) and event space (outcomes of interest).
One story about how probability theory was developed is that a gambler wanted to know when to bet more and when to bet less. He talked to friends of his who were mathematicians. Their names were Pierre de Fermat and Blaise Pascal. Since then, many other mathematicians have worked to develop probability theory.
Understanding probabilities is important in life. Examples of mundane questions that probability can answer are whether you need to carry an umbrella or wear a heavy coat on a given day. More important questions that probability can help with are the chances that a used car being purchased will need more maintenance. Other important examples are the chances of passing a class, the chances of winning the lottery, your chances of being in a car accident, and the chances that terrorists will attack the U.S.
Most people do not have a good understanding of probability, so they worry about being attacked by a terrorist but not about being in a car accident. The probability of being in a terrorist attack is much smaller than the probability of being in a car accident; thus, it actually would make more sense to worry about driving. Also, the chance of winning the lottery is small, yet many people will spend money on lottery tickets. If instead, they saved the money that they spend on the lottery, they would have more money. Events with a low probability (under 5%) are unlikely events. However, if an event has a high probability of happening (over 80%), then there is a good chance that the event will occur. This chapter will present some of the theories that you need to help make the determination of whether an event is likely to happen or not.
First, you need some definitions.
Experiment: an activity or process that will produce specific outcomes.
Outcomes: the result of an experiment.
Event: a set of certain outcomes of an experiment that you want to have happen.
Sample Space: a collection of all possible outcomes of the experiment. It is denoted as SS.
Event Space: the set of outcomes that make up an event. The symbol is usually a capital letter.
Start with an experiment. Suppose that the experiment is rolling a die. The sample space is {1, 2, 3, 4, 5, 6}. The event that you want is to get a 6, and the event space is {6}. To do this, roll a die 10 times. When you do that, you get a 6 two times. Based on this experiment, the probability of getting a 6 is 2 out of 10 or 1/5. To get more accuracy, repeat the experiment more times. It is easiest to put this in a table, where n represents the number of times the experiment is repeated. When you put the number of 6s found over the number of times you repeat the experiment, this is the relative frequency.
| n | Number of 6s | Relative Frequency |
|---|---|---|
| 10 | 2 | 0.2 |
| 50 | 6 | 0.12 |
| 100 | 18 | 0.18 |
| 500 | 81 | 0.162 |
| 1000 | 163 | 0.163 |
Notice that as n increases, the relative frequency approaches a number. It looks like it is approaching 0.163. You can say that the probability of getting a 6 is approximately 0.163. If you want more accuracy, then increase n even more.
These probabilities are called experimental (empirical) probabilities since they are found by experimentation. They come about from the relative frequencies and give an approximation of the true probability. The approximate probability of an event A, P(A), is
Experimental Probabilities
\(P(A)=\dfrac{\text { number of times } A \text { occurs }}{\text { number of times the experiment was repeated }}\)
For the event of getting a 6, the probability would by \(\dfrac{163}{1000}=0.163\).
Experimental probabilities are done whenever probabilities cannot be calculated using other means. For example, if you want to find the probability that a family has 5 children, you would look at many families and count how many have 5 children. Then you could calculate the probability. Another example is if you want to figure out if a die is fair. You would roll the die many times and count how often each side comes up. Make sure you repeat an experiment many times because otherwise, you won't be able to estimate the true probability. This is due to the law of large numbers.
Law of large numbers: as n increases, the relative frequency tends towards the actual probability value.
Probability, relative frequency, percentage, and proportion are all different words for the same concept. Also, probabilities can be given as percentages, decimals, or fractions.
Authors
"4.1: Empirical Probability" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY-SA 4.0
Attributions
"4.1: Empirical Probability" by Kathryn Kozak is licensed under CC BY-SA 4.0
Exercises
- The table below contains the number of M&M’s of each color that were found in a case (Madison, 2013). Find the probability of choosing each color based on this experiment. Round to 3 places.
Blue Brown Green Orange Red Yellow Total 481 371 483 544 372 369 2620 Table \(\PageIndex{2}\): M&M Distribution
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- Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made from January 1 to March 31. The table below gives the defect and the number of defects. Find the probability that the defect is caused by the following.
- It is scratched.
- Wrong height.
- Lost in the lab or wrong shape.
| Defect type | Number of defects |
|---|---|
| Scratch | 5865 |
| Right shaped - small | 4613 |
| Flaked | 1992 |
| Wrong axis | 1838 |
| Chamfer wrong | 1596 |
| Crazing, cracks | 1546 |
| Wrong shape | 1485 |
| Wrong PD | 1398 |
| Spots and bubbles | 1371 |
| Wrong height | 1130 |
| Right shape - big | 1105 |
| Lost in lab | 976 |
| Spots/bubble - intern | 976 |
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- In a math class of 45 students, 5 received an A, 11 received a B, and 20 received a C, and 8 . What is the probability that a student received the following grade? Round the answers to three decimal places.
- The student received an A.
- The student passed the class. The lowest grade is a C.
- The student did not pass.
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- At a local high school, the number of students with a class standing of freshman, sophomore, junior, and senior is listed in the table below. Compute the following probabilities. Round to two decimal places.
- The student is a freshman.
- The student is a sophomore or senior.
- The student is not a junior.
| Freshman | Sophomore | Junior | Senior |
|---|---|---|---|
| 901 | 755 | 697 | 552 |
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- Answers
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If you are an instructor and want the solutions to all the exercise questions for each section, please email Toros Berberyan.