4.1: Empirical Probability
- Page ID
- 45473
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- 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 of the earliest stories about probability comes from a gambler who wanted to know when to take risks and when to hold back. He turned to two mathematicians, Pierre de Fermat and Blaise Pascal, who began studying patterns in chance and uncertainty. Their work laid the foundation for what we now call probability, and many others have continued to develop these ideas over time.
Probability is useful in everyday life, even if we don’t always realize it. It can help answer simple questions like whether to bring an umbrella or wear a jacket. Still, it also helps with bigger decisions, such as understanding the risk of buying a used car, passing a class, or even being involved in a car accident. It can also explain why events like winning the lottery are so rare.
Many people tend to misunderstand probability. For example, some worry about rare events like terrorist attacks but don’t think much about more common risks like car accidents. In reality, the chance of being in a car accident is much higher. Similarly, even though the chances of winning the lottery are very small, people still spend money on tickets. Understanding probability helps us make more informed and rational decisions.
In general, events with a very low probability (less than 5%) are considered unlikely, while events with a high probability (greater than 80%) are likely to occur. This chapter will introduce key ideas that will help you better understand chance and make sense of uncertainty in everyday situations.
Key Definitions
Experiment: an activity or process that will produce specific outcomes.
Examples) Tossing one die, selecting a card from a standard deck, flipping 1 coin, etc.
Outcomes: the result of an experiment.
Example) When a die is flipped, it can lead to six possible outcomes: 1, 2, 3, 4, 5, or 6. These are the outcomes of the experiment.
Event: a set of certain outcomes of an experiment that you want to have happen.
Example) A die is tossed; one event can be getting an odd outcome. The possibilities are 1, 3, or 5.
Sample Space: a collection of all possible outcomes of the experiment. It is denoted as S.
Example) A coin is flipped; the sample space is S = {Heads, Tails}.
Event Space: the set of outcomes that make up an event. The symbol is usually a capital letter.
Example) A die is tossed; one event can be getting an odd outcome. The event space is E = {1, 3, 5}.
Visual Example of Definitions
Empirical Probability (Relative Frequency) Rule
Start with an experiment. Suppose we roll a die. The sample space is {1, 2, 3, 4, 5, 6}, and the event of interest is getting a 6, so the event space is {6}. Now roll the die 10 times. If a 6 appears 2 times, then the estimated probability of getting a 6 is 2 out of 10, or 1/5. To obtain a more accurate estimate, repeat the experiment many more times. It is helpful to organize the results in a table, where n represents the number of trials. The ratio of the number of 6s to the total number of trials is called 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 be \(\dfrac{163}{1000}=0.163\).
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.
Exercises
A student tracks how often they get a parking spot close to campus. Over 20 days, they find a close spot 6 times.
- What is the empirical probability of getting a close parking spot?'
- Based on this, how likely is it that they will find a close spot tomorrow?
- Answer
-
- Empirical probability = \( \dfrac{6}{20} = 0.30 \).
- There is a 30% chance of finding a close parking spot, so it is somewhat unlikely.
A coffee shop records customer drink choices during the morning rush. Out of 50 customers, 18 ordered iced coffee.
- What is the empirical probability that a randomly selected customer orders iced coffee?
- Would you consider this a common or uncommon choice?
- Answer
-
- Empirical probability = \( \dfrac{18}{50} = 0.36 \).
- There is a 36% chance a customer orders iced coffee, so it is a fairly common choice.
A student flips a coin 30 times and records 19 heads and 11 tails.
- What is the empirical probability of getting heads?
- How does this compare to the theoretical probability of heads?
- Answer
-
- Empirical probability of heads = \( \dfrac{19}{30} \approx 0.63 \).
- The theoretical probability is 0.50, so this result is higher than expected, but small samples can vary.
A student tracks how often they arrive at class on time. Over 25 class meetings, they arrive on time 20 times.
- What is the empirical probability of arriving at class on time?
- Would you consider this a common or uncommon choice?
- Answer
-
- Empirical probability = \( \dfrac{20}{25} = 0.80 \).
- There is an 80% chance the student arrives on time, so this is very likely.
A music app tracks how often a user skips songs. Out of 40 songs played, the user skips 10 songs.
- What is the empirical probability of a user skipping a song?
- Would you consider this a common or uncommon choice?
- Answer
-
- Empirical probability = \( \dfrac{10}{40} = 0.25 \).
- There is a 25% chance a song will be skipped, so skipping is somewhat uncommon.
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