Section 5.1: Probability Basics
- Page ID
- 215602
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- Compute theoretical probabilities
- Compute empirical probabilities
- The possible values a probability can take on
Introduction to Probability: Understanding the Foundation
Probability is the branch of mathematics that deals with uncertainty and chance. It is represented by a numerical measure on how likely something is to happen.
Every day, we encounter situations where we cannot predict exactly what will happen, but we can measure how likely different possibilities are. Whether we're wondering if it will rain, if our favorite team will win, or if we'll hit a red light on the way to work, we're thinking about probability. To study probability mathematically, we first need to understand four fundamental concepts that help us organize and analyze uncertain situations: experiments, outcomes, events, and sample spaces.
An experiment in probability is any process or activity that produces uncertain results.
Unlike scientific experiments that we might repeat to get the same result, probability experiments have unpredictable outcomes each time they're performed. Common examples of probability experiments include flipping a coin, rolling a die, drawing a card from a deck, or selecting a student randomly from a class. Even everyday activities like checking tomorrow's weather, choosing a route to work, or picking a movie to watch can be viewed as probability experiments. The key characteristic of any experiment is that we know what outcomes are possible, but we cannot predict with certainty which specific outcome will occur on any given trial.
The outcomes of an experiment are the individual results that can occur when the experiment is performed.
Each outcome represents one specific way the experiment can end. For example, when flipping a coin, there are exactly two possible outcomes: heads or tails. When rolling a standard six-sided die, the possible outcomes are 1, 2, 3, 4, 5, or 6. When drawing a single card from a standard deck, there are 52 possible outcomes (one for each card). It's important to note that outcomes should be defined clearly and specifically, and they should be mutually exclusive, meaning that only one outcome can occur at a time. Additionally, the outcomes should be equally likely when we're dealing with fair or unbiased experiments.
An event is a collection of one or more outcomes that we're interested in studying.
While an outcome is a single specific result, an event can include multiple outcomes that share some common characteristic. For instance, when rolling a die, "getting an even number" is an event that includes three outcomes: 2, 4, and 6. Similarly, "drawing a red card" from a deck is an event that includes 26 different outcomes (all hearts and diamonds). Events can be simple (containing just one outcome) or compound (containing multiple outcomes). We can also have events that include no outcomes (impossible events) or events that include all possible outcomes (certain events). Understanding events is crucial because probability questions typically ask about the likelihood of events occurring, not just individual outcomes.
The sample space is the complete set of all possible outcomes for a particular experiment.
It represents every single thing that could happen when the experiment is performed, with no possibilities left out. For a coin flip, the sample space is {heads, tails}. For rolling a die, the sample space is {1, 2, 3, 4, 5, 6}. For drawing a card from a standard deck, the sample space contains all 52 cards. The sample space serves as the foundation for calculating probabilities because it defines the "universe" of possibilities for that experiment. When we calculate the probability of an event, we're essentially comparing how many outcomes in the sample space belong to that event versus the total number of outcomes in the sample space.
Understanding experiments, outcomes, events, and sample spaces now gives us the vocabulary and conceptual tools needed to tackle more complex probability problems and make sense of uncertainty in our world. When we want to find the probability of an event, we first identify our experiment and determine its sample space. Then we identify which specific outcomes from the sample space make up the event we're interested in. After that, we use a basic probability formula to achieve the actual probability. This systematic approach allows us to move from informal thinking about chance to precise mathematical calculations. This is one of two main types of probability that we commonly study called a theoretical probability.
Theoretical probability is the probability calculated based on mathematical reasoning and the known structure of an experiment, assuming all outcomes are equally likely. It represents what we expect to happen in theory, without actually performing the experiment. The formula for theoretical probability is:
\[P(\text{Event }E)=\displaystyle\frac{\text{Number of Outcomes in }E}{\text{Number of Possible Outcomes in Sample Space}} \nonumber\]
Another type of probability is called an empirical probability.
Empirical probability (also called experimental probability) is the probability calculated based on actual experimental data or observed results. It represents what actually happens when an experiment is performed multiple times. The formula for empirical probability is:
\[P(\text{Event }E)=\displaystyle\frac{\text{Number of Times Event }E\text{ Occured}}{\text{Total Number of Trials}} \nonumber\]
The Fundamental Range
In probability theory, every probability value must fall within a specific, well-defined range. All probabilities are expressed as numbers between 0 and 1, inclusive. This means that for any event \(E\), we have \(0\leq E\leq 1\). This constraint is not arbitrary and it reflects the logical boundaries of what probability represents and provides a universal scale for measuring uncertainty.
Interpreting Values 0, 0.5, and 1
Probability = 0 (Impossible Events)
When \(P(E)=0\), the event is impossible and will never occur under the given conditions. This represents absolute certainty that something will not happen. For example, when rolling a standard six-sided die, the probability of rolling a 7 is 0/6 = 0, because 7 is not among the possible outcomes {1, 2, 3, 4, 5, 6}.
Probability = 0.5 (Equally Likely)
When \(P(E)=0.5\), the event E has an equal chance of occurring or not occurring. This represents maximum uncertainty. The classic example is flipping a fair coin. The probability of obtaining heads is 1/2 = 0.5. This means that over many trials, we expect heads to occur about half the time.
Probability = 1 (Certain Events)
When \(P(E)=1\), the event E is certain and will definitely occur every time the experiment is performed. This represents absolute certainty that something will happen. For instance, when rolling a standard die, the probability of rolling an odd or even number is 6/6 = 1, since every possible outcome satisfies this condition.
Interpreting Values Between 0 and 1
Low Probability (Close to 0)
Values near 0 indicate unlikely events. For example:
- \(P(E)=0.1\) means the event occurs about 10% of the time (1 in 10 trials)
- \(P(E)=0.05\) means the event occurs about 5% of the time (1 in 20 trials)
- These events are improbable but not impossible
Moderate Probability (Around 0.5)
Values near 0.5 indicate moderate uncertainty:
- \(P(E)=0.4\) means the event occurs about 40% of the time (4 in 10 trials)
- \(P(E)=0.6\) means the event occurs about 60% of the time (6 in 10 trials)
- These events have reasonable chances of occurring or not occurring
High Probability (Close to 1)
Values near 1 indicate likely events:
- \(P(E)=0.9\) means the event occurs about 90% of the time (9 in 10 trials)
- \(P(E)=0.95\) means the event occurs about 95% of the time (19 in 20 trials)
- These events are highly probable but not guaranteed
Common Misconceptions to Avoid
- Probabilities Cannot Exceed 1 - If your calculation yields a probability greater than 1, you've made an error. Check your counting of favorable outcomes and total outcomes.
- Probabilities Cannot Be Negative - Negative probabilities have no meaning in classical probability theory. If you get a negative result, review your calculation method.
The Range Applies to All Probability Types
Whether you're calculating theoretical probability, empirical probability, or any other type, the result must fall between 0 and 1. This universal constraint helps validate your calculations and interpretations.
Understanding the range of probability provides the foundation for all future work in statistics and probability theory. This scale gives us a common language for discussing uncertainty and helps ensure that our probability calculations and interpretations remain logically sound and practically meaningful.
Multiple Ways to Express Probabilities - Fractions, Decimals, and Percentages
The same probability can be expressed in different formats, all representing the same likelihood:
- Fraction: 1/4
- Decimal: 0.25
- Percentage: 25%
All three expressions represent the same probability and fall within our required range of 0 to 1, inclusive (when using decimal form).
Decide whether or not each given number could represent a probability.
- \(0.56\)
- \(84\%\)
- \(\frac{7}{9}\)
- \(\frac{5}{4}\)
- \(0\)
- \(-0.48\)
- \(62\%\)
- \(2\)
- \(-\frac{1}{3}\)
- \(110\%\)
- \(\frac{5}{5}\)
- \(1.79\)
- \(-25\)
✅ Solution:
So, we are looking for numbers that are between \(0\) and \(1\) inclusive.
- Yes, \(0.56\) could represent a probability; \(0.56\) is between \(0\) and \(1\) inclusive.
- Yes, \(84\%\) could represent a probability; \(84\%=0.84\).
- Yes, \(\frac{7}{9}\) could represent a probability; \(\frac{7}{9} \approx 0.778\); \(0.778\) is between \(0\) and \(1\) inclusive.
- No, \(\frac{5}{4}\) cannot represent a probability; \(\frac{5}{4}=1.25\); \(1.25\) is larger than \(1\).
- Yes, \(0\) could represent a probability; \(0\) is between \(0\) and \(1\) inclusive.
- No, \(-0.48\) cannot represent a probability; negative numbers cannot represent probabilities.
- Yes, \(62\%\) could represent a probability; \(62\%=0.62\).
- No, \(2\) cannot represent a probability; \(2\) is larger than \(1\).
- No, \(-\frac{1}{3}\) cannot represent a probability; negative numbers cannot represent probabilities.
- No, \(110\%\) cannot represent a probability; \(110\%=1.1\) ; \(1.1\) is larger than \(1\).
- Yes, \(\frac{5}{5}\) could represent a probability; \(\frac{5}{5}=1\); \(1\) is between \(0\) and \(1\) inclusive.
- No, \(1.79\) cannot represent a probability; \(1.79\) is larger than \(1\).
- No, \(-25\) cannot represent a probability; negative numbers cannot represent probabilities.
A single six-sided die is rolled. Determine the probability of each of the following outcomes.
- a '\(6\)'
- odd number
- a number less than '\(3\)'
- a number greater than or equal to '\(4\)'
- a number less than '\(7\)'
- a number greater than '\(9\)'
✅ Solution:
There are 6 outcomes in the sample space, which is {\(1, 2, 3, 4, 5, 6\)}.
- \(\frac{1}{6}\); there is one outcome '\(6\)' from six total outcomes.
- \(\frac{3}{6}=\frac{1}{2}\); there are three odd outcomes '\(1\)', '\(3\)', and '\(5\)' from the six total outcomes.
- \(\frac{2}{6}=\frac{1}{3}\); there are two outcomes '\(1\)', and '\(2\)' from the six total outcomes.
- \(\frac{3}{6}=\frac{1}{2}\); there are three outcomes '\(4\)', '\(5\)', and '\(6\)' from the six total outcomes.
- \(\frac{6}{6}=1\); all six total outcomes are already less than \(7\).
- \(\frac{0}{6}=0\); there are no outcomes greater than \(9\) from the six total outcomes.
The Weather Channel App says that the chance of rain in your area is \(30\%\). What is the probability of rain? Express your answer as a percentage and a decimal.
✅ Solution:
\(0.3\) or \(30\%\)
According to the American Veterinary Medical Association (AVMA), as of 2024, the proportion of U.S. households who owned a dog is 0.455. What is the probability that a randomly selected house owns a dog?
✅ Solution:
\(0.455\) or \(45.5\%\)
A marble is drawn randomly from a jar that contains \(6\) red marbles, \(2\) white marbles, and \(5\) blue marbles. Find the probability of each given event.
- A red marble is drawn
- A white marble is drawn
- A green marble is drawn
- A blue marble is drawn
- A red or blue marble is drawn
✅ Solution:
- \(\frac{6}{13}\); there are \(6\) red marbles from a total of \(13\) marbles.
- \(\frac{2}{13}\); there are \(2\) white marbles from a total of \(13\) marbles.
- \(\frac{0}{13}=0\); there are no green marbles from a total of \(13\) marbles.
- \(\frac{5}{13}\); there are \(5\) blue marbles from a total of \(13\) marbles.
- \(\frac{11}{13}\); there are \(6\) red marbles and \(5\) blue marbles. So there are \(11\) red or blue marbles from a total of \(13\) marbles.
In a survey, 160 people indicated they prefer cats, 225 indicated they prefer dogs, and 40 indicated they don’t enjoy either pet. Find the probability that if a person is chosen at random, they prefer cats.
✅ Solution:
First, we need to add all of the people in the survey, \(160+225+40=425\).
So, the probability of a person preferring cats is \(\frac{160}{425}=0.37647058823 \approx 0.376\).
A spinner has sectors labeled A, B, C, and D with equal probability. The spinner gets spun.
- What is the probability that the spinner lands on C?
- What is the probability that the spinner does not land on B?
- What is the probability that the spinner lands on A or D?
✅ Solution:
Since there are four sectors all with equal probability, then the probability 1/4 represents each sector.
- \(\frac{1}{4}\); sector C is one of a total of 4 sectors.
- \(\frac{3}{4}\); not B, so sectors A, C, and D are three of a total of 4 sectors..
- \(\frac{2}{4}=\frac{1}{2}\); sectors A and D are two of a total of 4 sectors.
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Decide whether or not each given number could represent a probability.
- \(0.23\)
- \(\frac{3}{4}\)
- \(-0.65\)
- \(120\%\)
- \(\frac{3}{2}\)
- \(48\)
- \(62\%\)
- \(1.5\)
- \(0\)
- \(0.5\)
- \(1\)
- \(0.15\)
- \(\frac{4}{4}\)
- \(-\frac{2}{5}\)
- A single six-sided die is rolled. Determine the probability of each of the following outcomes.
- a '\(1\)'
- an even number
- a number less than '\(2\)'
- a number greater than or equal to '\(5\)'
- a number less than '\(1\)'
- a number greater than '\(0\)'
- In a class of \(30\) students, \(12\) are biology majors. If one student is chosen at random, what is the probability the student is a biology major?
- A fair coin is tossed. What is the probability of getting tails?
- A jar contains \(5\) red marbles, \(7\) blue marbles, and \(8\) green marbles.
- What is the probability of selecting a blue marble?
- What is the probability of selecting a green marble?
- What is the probability of selecting a red or a green marble?
- What is the probability of selecting a purple marble?
- What is the probability of selecting a red and a blue marble?
- A spinner has sectors labeled A, B, C, D, and E with equal probability. The spinner is spun.
- What is the probability that the spinner lands on B?
- What is the probability that the spinner does not land on D?
- What is the probability that the spinner lands on A or E?
- A study finds that \(1\) out of every \(20\) phone batteries fails within the first year. If you choose a phone at random from that model, what is the probability its battery will not fail in the first year?
- Answers
-
- a) Yes; b) Yes; c) No; d) No; e) No; f) No; g) Yes; h) No; i) Yes; j) Yes; k) Yes; l) Yes; m) Yes; n) No.
- a) \(\frac{1}{6}\); b) \(\frac{3}{6}=\frac{1}{2}\); c) \(\frac{1}{6}\); d) \(\frac{2}{6}=\frac{1}{3}\); e) \(\frac{0}{6}=0\); f) \(\frac{6}{6}=1\).
- \(\frac{12}{30}=\frac{2}{5}\)
- \(\frac{1}{2}\)
- a) \(\frac{7}{20}\); b) \(\frac{8}{20}=\frac{2}{5}\); c) \(\frac{13}{20}\); d) \(\frac{0}{20}=0\); e) \(\frac{12}{20}=\frac{3}{5}\).
- a) \(\frac{1}{5}\); b) \(\frac{4}{5}\); c) \(\frac{2}{5}\).
- \(\frac{19}{20}\)