9.5.1: Independent Events (Exercises)
- Page ID
- 201187
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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 distribution of the number of fiction and non-fiction books checked out at a city's main library and at a smaller branch on a given day is as follows.
| MAIN (M) | BRANCH (B) | TOTAL | |
|---|---|---|---|
| FICTION (F) | 300 | 100 | 400 |
| NON-FICTION (N) | 150 | 50 | 200 |
| TOTALS | 450 | 150 | 600 |
Use this table to determine the following probabilities:
- \(P(F)\)
- \(P(M | F)\)
- \(P(N | B)\)
- Is the fact that a person checks out a fiction book independent of the main library? Use probabilities to justify your conclusion.
For a two-child family, let the events \(E\), \(F\), and \(G\) be as follows.
\(E\): The family has at least one boy
\(F\): The family has children of both sexes
\(G\): The family's first born is a boy
- Find the following.
- \(P(E)\)
- \(P(F)\)
- \(P(E \cap F)\)
- Are \(E\) and \(F\) independent? Use probabilities to justify your conclusion.
- Find the following.
- \(P(F)\)
- \(P(G)\)
- \(P(F \cap G)\)
- Are \(F\) and \(G\) independent? Use probabilities to justify your conclusion.
Do the following problems involving independence.
- If \(P(E) = .6\), \(P(F) = .2\), and \(E\) and \(F\) are independent, find \(P\)(\(E\) and \(F\)).
- If \(P(E) = .6\), \(P(F) = .2\), and \(E\) and \(F\) are independent, find \(P\)(\(E\) or \(F\)).
- If \(P(E) = .9\), \(P(F | E) = .36\), and \(E\) and \(F\) are independent, find \(P(F)\).
- If \(P(E) = .6\), \(P\)(\(E\) or \(F\)) = .8, and \(E\) and \(F\) are independent, find \(P(F)\).
- In a survey of 100 people, 40 were casual drinkers, and 60 did not drink. Of the ones who drank, 6 had minor headaches. Of the non-drinkers, 9 had minor headaches. Are the events "drinkers" and "had headaches" independent?
- It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
- John's probability of passing statistics is 40%, and Linda's probability of passing the same course is 70%. If the two events are independent, find the following probabilities.
- \(P\)( both of them will pass statistics)
- \(P\)(at least one of them will pass statistics)
- Jane is flying home for the Christmas holidays. She has to change planes twice. There is an 80% chance that she will make the first connection, and a 90% chance that she will make the second connection. If the two events are independent, find the probabilities:
- \(P\)( Jane will make both connections)
- \(P\)(Jane will make at least one connection)
For a three-child family, let the events \(E\), \(F\), and \(G\) be as follows.
\(E\): The family has at least one boy
\(F\): The family has children of both sexes
\(G\): The family's first born is a boy
- Find the following.
- \(P(E)\)
- \(P(F)\)
- \(P(E \cap F)\)
- Are \(E\) and \(F\) independent?
- Find the following.
- \(P(F)\)
- \(P(G)\)
- \(P(F \cap G)\)
- Are \(F\) and \(G\) independent?
Answer the following:
- \(P(K|D) = 0.7\), \(P(D) = 0.25\) and \(P(K)=0.7\)
- Are events \(K\) and \(D\) independent? Use probabilities to justify your conclusion.
- Find \(P(K \cap D)\)
- \(P(R|S) = 0.4\), \(P(S) = 0.2\) and \(P(R)=0.3\)
- Are events \(R\) and \(S\) independent? Use probabilities to justify your conclusion.
- Find \(P(R \cap S)\)
- At a college: 54% of students are female, 25% of students are majoring in engineering, 5% of female students are majoring in engineering.
Event \(E\) = student is majoring in engineering. Event \(F\) = student is female.- Are events \(E\) and \(F\) independent? Use probabilities to justify your conclusion.
- Find \(P(E \cap F)\)
- At a college: 54% of all students are female, 60% of all students receive financial aid, 60% of female students receive financial aid.
Event \(A\) = student receives financial aid. Event \(F\) = student is female- Are events \(A\) and \(F\) independent? Use probabilities to justify your conclusion.
- Find \(P(A \cap F)\)