4.1: What is Probability
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In our everyday lives, we’re constantly working with uncertainty. Will it rain today? What are the chances of getting in a car accident? How confident can we be that a medicine works? This is where probability comes in.
Probability is the critical for developing statistics. It gives us a language and a set of tools for understanding, describing, and modeling randomness. It allows us to measure uncertainty and reason logically about outcomes that aren’t predictable with certainty.
Definition: Probability
Probability is the long-run relative frequency of an event. This is the proportion of times an outcome would occur if we repeated a random process over and over, under the same conditions.
A Simple Example: Coin Flipping
Think about flipping a coin. If you flip it once, you might get heads. If you flip it twice, maybe tails. But if you flip the coin 100 times, or 1,000 times, you’ll notice a pattern emerge: you get heads about half the time.
That’s the idea behind probability. In the long run, there is a predictable pattern in a repeating random process, even though you can't predict what will happen in the short run.
Important Concept: Probability Is Not Prediction
Probability doesn’t guarantee an outcome, as it only describes what tends to happen in many repetitions. Even though flipping a coin has a 50% chance of heads, we might still get 7 heads out of 10 flips, or even 4 heads out of 10. That’s not unusual in the short run. It is impossible for us to use probability to predict the next flip, only know how what we predict the statistics from many many flips to be.
As we gather more data or observe more repetitions, the pattern in the coin stabilizes. That’s why probability is often described as measuring long-term relative frequency.
Explore It Yourself: Probability in the Long Run
To better understand how probability behaves in the long run, try the interactive applet below.
In this applet:
- You can set the value of the theoretical probability \( P_{success} \) (e.g., 0.5 for a fair coin).
- The word trials refers to repeated investigations of a probabilistic idea. We represent the trials as \( k \) and you can see the effect of it by pressing play.
- As you increase \( k \), the graph will show the proportion of successes over trials based on random outcomes using that value of \( P \).
When we have a small number of trials, small deviations from a 50/50 ratio stand out more. For example, if we flip 4 coins and get 3 heads, that is 75% of our data set, however it is only one value off from 50/50. If we flip 100 coins, being off by one would be 51 heads to 49 tails. This is very close to 50/50!
Think About It:
Try running the app with \( P = 0.5 \), \( k = 50 \), and then again with \( k = 1000 \).
What does the graph look like when trials are small vs. large? How does this demonstrate the definition of probability as a long-run relative frequency?
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What’s Coming Next
In the next sections, we’ll explore tools that help us count and calculate probabilities. We’ll look at defining sample spaces, building rules for combining probabilities, and simulating some events to get a better feel for what random processes look like.
For now, remember: probability is not a prediction engine; it’s a map of what we expect over time.