4.2: The Geometric Distribution
- Page ID
- 48785
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)There are many probability experiments where a trial has only two outcomes. For example, asking a group of individuals if they vote yes on a proposition, or randomly guessing on a multiple choice test. When we conduct a sequence of independent trials with only two outcomes per trial, we are conducting a binomial experiment.
- Which of the following only have two possible outcomes:
- Rolling a 4 on a 6-sided die
- Examining the global temperature change over time.
- Measuring the height of adult in California
- Meeting a person that is infected with Covid-19
- Rolling a 4 on a 6-sided die
Characteristics of a Geometric Experiment
A geometric experiment is a probability experiment with the following characteristics:
- Each trial has exactly two possible outcomes which are labeled success and failure.
- The probability of success is the same for each trial. We denote the probability of success as p and the probability of failure as \(q=1-p\).
- We look for when the first and only success occurs. There must be at least one trial, and in theory, we could repeat trials forever.
- Go to https://www.random.org/dice/ and roll 1 die. Roll the die counting the number of trials it took to roll a 5. Keep track of your rolls in the table below.
Tally
On what attempt did you succeed in rolling a 5?
- Assume we will roll a fair six-sided die.
- What is the probability of rolling a 5? We define rolling a 5 as success, and therefore, we are computing the probability of success.
- What is the probability that we will not roll a 5? Use the complement rule to compute the probability of failure.
- What is the probability of rolling a 5? We define rolling a 5 as success, and therefore, we are computing the probability of success.
- Suppose we are rolling a fair six-sided die.
- What is the probability that we will roll a 5 (succeed) on the first attempt?
- What is the probability that we will roll a 5 (succeed) on the second attempt? In this case, we fail on the first try and succeed on the second try. Use the multiplication rule for independent events.
- What is the probability that we will roll a 5 (succeed) on the first attempt?
- Suppose we are rolling a fair six-sided die.
- What is the probability that we will roll a die and succeed (roll a 5) on the third attempt? In this case, we fail on the first and second tries and succeed on the third try. Use the multiplication rule for independent events.
- What is the probability that we will roll a die and succeed (roll a 5) on the fourth attempt? In this case, we fail on the first and second and third tries and succeed on the fourth try. Use the multiplication rule for independent events.
- What is the probability that we will roll a die and succeed (roll a 5) on the third attempt? In this case, we fail on the first and second tries and succeed on the third try. Use the multiplication rule for independent events.
- What is the formula for computing geometric probability? So far, this is what we have come up with:
\[\begin{aligned}
& P(1)=\frac{1}{6} \\
& P(2)=\frac{5}{6} \cdot \frac{1}{6} \\
& P(3)=\frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6}=\left(\frac{5}{6}\right)^2 \cdot \frac{1}{6} \\
& P(4)=\frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6}=\left(\frac{5}{6}\right)^3 \cdot \frac{1}{6}
\end{aligned}\]What patterns do you notice?
- What is P(5)?
Geometric Probability
In general, the probability of succeeding only once on the \(x\)th attempt is
\[P(x)=q^{x-1} p\nonumber\]
where \(p\) is the probability of success and \(q=1-p\) is the probability of failure.
- You try! You enter a darts tournament. The probability of hitting the bullseye is 17%. What is the probability that you hit the bullseye on the 7th attempt? You can upload an image to show your thinking.