8: 7. Probability without Equally Likely Outcomes
- Page ID
- 25693
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The following topics are included in this series of eight videos.
- Not Equally Likely, Example 1
- Not Equally Likely, Example 2
- Not Equally Likely, Example 3
- Disjoint Events
- Not Equally Likely, Example 4
- Not Equally Likely, Example 5
- Not Equally Likely, Example 6
- Not Equally Likely, Example 7
Prework:
- Suppose there are some orange, red, and purple balls in a bag, but you don’t know how many of each. You reach in and grab one. If the chance of picking an orange is \(25\%\) and the chance of picking a red is \(37\%\), what is the probability that you pick either an orange or purple ball?
- Let \(S\) be the sample space for an experiment. Let \(S = \{O_1, O_2, O_3, O_4, O_5\}\). Assume that \(Pr[O_1] = 0.1\) and \(Pr[O_4] = 0.25\). Assume that \(O_2\) and \(O_3\) are equally likely and that \(O_5\) is three times as likely as \(O_2\). Find \(Pr[O_2]\) and \(Pr[O_5]\).
- Suppose that \(Pr[A] = 0.4\), \(Pr[B] = 0.6\), and \(Pr[A \cap B^c] = 0.16\). Determine \(Pr[A^c], Pr[A\cup B], Pr[A^c\cap B]\).
Solutions:
- Since the only three outcomes of this experiment are picking an orange, red, or purple ball, we have a \(100-25-37=38\%\) chance of picking a purple ball. Therefore \(Pr(\text{orange or purple})=Pr(\text{orange})+Pr(\text{purple})=0.25+0.38=0.63.\)
- Because there are only these 5 outcomes, we know \(Pr(O_1)+Pr(O_2)+Pr(O_3)+Pr(O_4)+Pr(O_5)=1\). Since \(O_2\) and \(O_3\) are equally likely, we know that \(Pr(O_3)=Pr(O_2)\), and since \(O_5\) is three times as likely as \(O_2\), we know \(Pr(O_5)=3Pr(O_2)\). Substituting this information and the given probabilities into our initial equation, we get \(0.1+Pr(O_2)+Pr(O_2)+0.25+3Pr(O_2)=1\). Combining like terms we see that \(0.35+5Pr(O_2)=1\). We subtract \(0.35\) and divide by \(5\) to get that \(Pr(O_2)=0.13\), and therefore \(Pr(O_5)=3Pr(O_2)=3\cdot 0.13=0.39.\)
- The answer to the first question is that \(Pr(A^c)=0.6\) since, for any set \(A\), \(Pr(A)+Pr(A^c)=1\). We next make the Venn diagram shown below by first using \(Pr(A\cap B^c)=0.16.\) By subtraction, \(Pr(A\cap B)=0.24\) and hence \(Pr(A^c\cap B)=0.36.\) Therefore, \(Pr(A^c\cap B^c)=0.24.\) Using the diagram we see that \(Pr(A\cup B)=0.16+0.24+0.36=0.76\) and \(Pr(A^c\cap B)=0.36\).
