5.E: Basic Concepts of Probability (Exercises)
- Page ID
- 4920
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\): Counting
A typical PIN is a sequence of any seven symbols chosen from the \(26\) letters in the alphabets and the ten digits \(0, 1,....,9\) with repetition allowed.
- How many PIN's are available?
- Supposing that symbols cannot be repeated, then how many PIN's are available?
Exercise \(\PageIndex{2}\): Counting
How many (Canadian) postal codes would be possible without the repetition of letters or numbers?
Exercise \(\PageIndex{3}\): Counting
Seven women and nine men are on the faculty in the Mathematics Department.
- How many different committees are there which are made up of five members of the department, at least one of which is a woman?
- How many ways can the five-person committee be arranged around a circular table?
Exercise \(\PageIndex{4}\): Probability
You have 23 people in a room. What is the probability that the two of them have the same birthday?
Exercise \(\PageIndex{5}\):Probability
A coin is tossed \(7\) times, find
- the probability that we see at least \(2\) heads?
- the probability that we see exactly \(2\) heads?
- the probability that we see exactly \(2\) heads or exactly \(4\) heads?
Exercise \(\PageIndex{6}\):Probability
What is the conditional probability that the sum of the dice is \(10, 11,\) or \(12\), given that the first die rolled comes up a \(6\)?
Exercise \(\PageIndex{7}\):Probability
We have a standard deck of \(52\) cards, and you select a card randomly.
- What is the probability you select a \(5\)?
- What is the probability you don’t select a \(5\)?
- What is the probability you select a \(5\) or \(5\)? Are these mutually exclusive events or not?
- What is the probability you select a \(5\) or a diamond? Are these mutually exclusive events or not?
- What is the probability you will select the Jack of Spades, given that you have selected a face card?
- What is the probability you select a diamond given you have selected a red card?
- What is the probability you selected a red card, given you’ve selected a diamond?
Now, suppose you select two cards from the deck.
- What is the probability you select two Jacks, assuming you replace a card and reshuffle the deck before selecting again? Are these dependent or independent events?
- What is the probability you select two Jacks, assuming you don’t replace a card once selected? Are these dependent or independent events?
- What is the probability you will select a Jack and an Ace – again assuming replacement? Careful- you could select the Jack, then the Ace, or the Ace, then the Jack.
Exercise \(\PageIndex{8}\): Probability
In a town of \(10,000 \) people, \(400\) have beards (all men), \(4000\) are adult men, and \(5\) of the townspeople are murderers. All five murderers are men, and four of the murderers have beards.
Suppose you go to this town and select a towns person at random.
- Let \(A\) be the event that the person turns out to be one of the five murderers.
- Let \(B\) be the event the person is bearded.
- Let \(C\) be the event the person is an adult male.
Find \(P(A), P(A \, given \, B), P(B \, given \, A), P(A \, given \,not\, B), P(A \, given \, C), P(A \, given \,not\, C).\)
Exercise \(\PageIndex{9}\): False Positive
Disease X affects about \(1\) percent of the population. A test for Disease X will test positive on all afflicted with the disease and will also test positive for \(5%\) of the population who do not have the disease.
- What is the probability a randomly chosen person has disease X?
- What is the probability a randomly chosen person will test positive for the disease?
- Suppose you test positive for the disease. What is the probability you don’t actually have the disease? (This is the conditional probability that you don’t have the disease given you tested positive for it).
- What would be the probability that you test positive for the disease twice, given that you don’t have disease X?
- Can you identify (at least 2) criticism’s of our theoretical probability calculations here?
Exercise \(\PageIndex{10}\): Probability
- the probability that we see at least \(2\) heads?
- the probability that we see exactly \(2\) heads?
- the probability that we see exactly \( 2 \) heads or exactly \( 4 \) heads?
Exercise \(\PageIndex{11}\): Probability
A multiple-choice test has ten questions, each with four possible answers. A student guesses all ten questions.
- Find the probability that the student will get all ten questions right.
- Find the probability that the student will get at least one question right.
Exercise \(\PageIndex{12}\): Probability
Suppose two fair dice are thrown.
- What is the probability the sum of the dice is at most \(4\)?
- What is the probability the sum of the dice is more than \(4\)?
- What are the odds the sum of the dice is at most \(4\)?
- What is the probability the sum of the dice is at most \(4\), given that the first die shows a \(3\)?
Exercise \(\PageIndex{13}\): Combination
If \(n \geq k+2\) and \(k \geq 2\), show that \( {n \choose k } - {n-2 \choose k} - {n-2 \choose k-2}\) is even.
Exercise \(\PageIndex{14}\): Combination
Find the coefficient of \(x^{5}\) in the binomial expansion of
\[\left( \frac{2}{x}+x^2 \right)^{25}\].
Exercise \(\PageIndex{15}\): Combination
For natural numbers \(n\) and \(r\), \(r<n\), show that
\[\displaystyle {n+1 \choose r } = {n \choose r} + {n \choose r-1}. \]
Exercise \(\PageIndex{16}\):
A fair coin, a double-headed coin and a double-tailed coin are placed in a bag. A coin is randomly selected. The coin selected is then tossed.
1. Find the probability that the coin lands with a “head”.
2. When the coin is tossed, it lands “tail”. Find the probability that it is a double-tailed coin.
- Answer
-
\(\dfrac{1}{2} \dfrac{3}{4}\).