Chapter 12

\(S_{n}\) is binomially distributed with parameters \(n\) and \(p\). For \(n=100\) and \(p=0.01\), compute \(P\left(S_{n}=0\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Let \(X\) and \(Y\) be two random variables with the following joint distribution: $$ \begin{array}{ccc} \hline & \boldsymbol{X}=\mathbf{0} & \boldsymbol{X}=\mathbf{1} \\ \hline \boldsymbol{Y}=\mathbf{0} & 0.3 & 0.1 \\ \boldsymbol{Y}=\mathbf{1} & 0.2 & 0.4 \\ \hline \end{array} $$ (a) Find \(P(X=1, Y=0)\). (b) Find \(P(X=1)\). (c) Find \(P(Y=0)\). (d) Find \(P(Y=0 \mid X=1)\).

Suppose the weight of an animal is normally distributed with mean \(3720 \mathrm{~g}\) and standard deviation \(527 \mathrm{~g} .\) What percentage of the population has a weight that exceeds \(5000 \mathrm{~g}\) ?

Roll two fair dice, one after the other, and find the probability that the first number is larger than the second number.

You are dealt 1 card from a standard deck of 52 cards. If \(A\) denotes the event that the card is a spade and if \(B\) denotes the event that the card is an ace, determine whether \(A\) and \(B\) are independent.

Amin owns a 4-GB music storage device that holds 1000 songs. How many different playlists of 20 songs are there if the order of the songs is important?

fit a linear regression line through the given points and compute the coefficient of determination. \((0,0.1),(1,-1.3),(2,-3.5),(3,-5.7),(4,-5.8)\)

\(S_{n}\) is binomially distributed with parameters \(n\) and \(p\). For \(n=100\) and \(p=0.1\), compute \(P\left(S_{n}=10\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Let \(X\) and \(Y\) be two random variables with the following joint distribution: $$ \begin{array}{ccc} \hline & X=0 & X=1 \\ \hline \boldsymbol{Y}=\mathbf{0} & 0.2 & 0.0 \\ \boldsymbol{Y}=\mathbf{1} & 0.3 & 0.5 \\ \hline \end{array} $$ (a) Find \(P(X=0, Y=1)\). (b) Find \(P(X=0)\). (c) Find \(P(Y=1)\). (d) Find \(P(X=0 \mid Y=0)\).

Suppose the height of an adult animal is normally distributed with mean \(17.2\) in. Find the standard deviation if \(10 \%\) of the animals have a height that exceeds 19 in.

Roll two fair dice and find the probability that the minimum of the two numbers will be greater than 4.

You are dealt 2 cards from a standard deck of 52 cards. If \(A\) denotes the event that the first card is an ace and \(B\) denotes the event that the second card is an ace, determine whether \(A\) and \(B\) are independent.

A bookstore has 300 science fiction books. Molly wants to buy 5 of the 300 science fiction books. How many selections are there?

Show that the sum of the residuals about any linear regression line is equal to 0 .

\(S_{n}\) is binomially distributed with parameters \(n\) and \(p\). For \(n=50\) and \(p=0.1\), compute \(P\left(S_{n}=5\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Let \(X\) and \(Y\) be two independent random variables with probability mass function described by the following table: $$ \begin{array}{ccc} \hline \boldsymbol{k} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{k}) & \boldsymbol{P}(\boldsymbol{Y}=\boldsymbol{k}) \\ \hline-2 & 0.1 & 0.2 \\ -1 & 0 & 0.2 \\ 0 & 0.3 & 0.1 \\ 1 & 0.4 & 0.3 \\ 2 & 0.05 & 0 \\ 3 & 0.15 & 0.2 \\ \hline \end{array} $$ (a) Find \(E(X)\) and \(E(Y)\). (b) Find \(E(X+Y)\). (c) Find \(\operatorname{var}(X)\) and \(\operatorname{var}(Y)\). (d) Find \(\operatorname{var}(X+Y)\).

Suppose that \(X\) is normally distributed with mean 2 and standard deviation \(1 .\) Find \(P(0 \leq X \leq 3)\).

An urn contains five green and six blue balls. You take two balls out of the urn, one after the other, without replacement. If \(A\) denotes the event that the first ball is green and \(B\) denotes the event that the second ball is green, determine whether \(A\) and \(B\) are independent.

A box contains five red and four blue balls. You choose two balls. (a) How many possible selections contain exactly two red balls, how many exactly two blue balls, and how many exactly one of each color? (b) Show that the sum of the number of choices for the three cases in (a) is equal to the number of ways that you can select two balls out of the nine balls in the box.

\(S_{n}\) is binomially distributed with parameters \(n\) and \(p\). For \(n=50\) and \(p=0.5\), compute \(P\left(S_{n}=25\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Let \(X\) and \(Y\) be two independent random variables with probability mass function described by the following table: $$ \begin{array}{rcc} \hline \boldsymbol{k} & \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{k}) & \boldsymbol{P}(\boldsymbol{Y}=\boldsymbol{k}) \\ \hline-3 & 0.1 & 0.1 \\ -1 & 0.1 & 0.2 \\ 0 & 0.2 & 0.1 \\ 0.5 & 0.3 & 0.3 \\ 2 & 0.15 & 0.1 \\ 2.5 & 0.15 & 0.2 \\ \hline \end{array} $$ (a) Find \(E(X)\) and \(E(Y)\). (b) Find \(E(X+Y)\). (c) Find \(\operatorname{var}(X)\) and \(\operatorname{var}(Y)\). (d) Find \(\operatorname{var}(X+Y)\).

Suppose that \(X\) is normally distributed with mean \(-1\) and standard deviation 2. Find \(P(-3.5 \leq X \leq 0.5)\).

An urn contains four green and three blue balls. You take one ball out of the urn, note its color, and replace it. You then take a second ball out of the urn, note its color, and replace it. If \(A\) denotes the event that the first ball is green and \(B\) denotes the event that the second ball is green, determine whether \(A\) and \(B\) are independent.

Twelve children are divided up into three groups, of five, four, and three children, respectively. In how many ways can this be done if the order within each group is not important?

Suppose you want to estimate the proportion of people in the United States who do not believe in evolution. You happen to take a class on evolutionary theory at a U.S. college that is attended by 200 students, all of whom are biology majors. Do you think you would get an accurate estimate if you asked all 200 students in vour class? Discuss.

We have two formulas for computing the variance of \(X\), namely, $$ \operatorname{var}(X)=E\left[(X-E(X))^{2}\right] $$ and $$ \operatorname{var}(X)=E\left(X^{2}\right)-[E(X)]^{2} $$ (a) Explain why \(\operatorname{var}(X) \geq 0\). (b) Use your results in (a) to explain why $$ E\left(X^{2}\right) \geq[E(X)]^{2} $$

Suppose that \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Show that \(E(X)=\mu .\) [You may use the fact that if \(Z\) is standard normally distributed, then \(E(Z)=0\) and \(\operatorname{var}(X)=1 .]\)

Suppose that two parents are of genotype \(A a\). What is the probability that their offspring is of genotype \(A a ?\) (Assume Mendel's first law.)

Assume a \(1: 1\) sex ratio. A family has three children. Find the probability of each event: (a) \(A=\\{\) all children are girls \(\\}\) (b) \(B=\\{\) at least one boy \(\\}\) (c) \(C=\\{\) at least two girls \(\\}\) (d) \(D=\\{\) at most two boys \(\\}\)

Five A's, three B's, and six C's are to be arranged into a 14 letter "word". How many different words can you form?

A soft-drink company introduces a new beverage. One month later, the company wants to know whether its marketing strategies have reached young adults of ages \(18-20 .\) You happen to work part time for the marketing company that is conducting the survey. At the same time, you are taking a calculus class that is attended by 250 students. It would be easy for you to hand out a survey in class. Would you suggest this to your supervisor in the marketing company? Discuss.

Assume that \(X\) is a discrete random variable with finite range. Show that if \(\operatorname{var}(X)=0\), then \(P(X=E(X))=1\).

Suppose that \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Show that \(\operatorname{var}(X)=\sigma^{2} .\) [You may use the fact that if \(Z\) is standard normally distributed, then \(E(Z)=0\) and \(\operatorname{var}(X)=1 .]\)

Suppose that one parent is of genotype \(A A\) and the other is of genotype \(A a .\) What is the probability that their offspring is of genotype \(A A ?\) (Assume Mendel's first law.)

Assume that \(20 \%\) of a very common insect species in your study area is parasitized. Assume that insects are parasitized independently of each other. If you collect 10 specimens of this species, what is the probability that no more than 2 specimens in your sample are parasitized?

A bag contains 45 beans of three different varieties. Each variety is represented 15 times in the bag. You grab 9 beans out of the bag. (a) Count the number of ways that each variety can be represented exactly three times in your sample. (b) Count the number of ways that only one variety appears in your sample.

Clementine oranges are sold in boxes. Each box contains 50 clementines. The probability that a clementine in a box is spoiled is \(0.01\) (a) Use an appropriate approximation to determine the probability that a box contains 0,1, or at least 2 spoiled clementines. (b) A shipment of clementines (said to be hybrid crossings between oranges and tangerines) with 100 boxes is considered unacceptable if \(35 \%\) or more of the boxes contain spoiled clementines. What is the probability that a shipment is unacceptable?

Toss a fair coin 10 times. Let \(X\) be the number of heads. Find (a) \(P(X=5)\). (b) \(P(X \geq 8)\). (c) \(P(X \leq 9)\).

Suppose that \(X\) is standard normally distributed. Find \(E(|X|)\).

A family has three children. Assuming a \(1: 1\) sex ratio, what is the probability that all of the children are girls?

A multiple-choice question has four choices, and a test has a total of 10 multiple-choice questions. A student passes the test only if he or she answers all questions correctly. If the student guesses the answers to all questions randomly, find the probability that he or she will pass.

Let \(S=\\{a, b, c\\} .\) List all possible subsets, and argue that the total number of subsets is \(2^{3}=8\).

Turner's Syndrome Turner's syndrome is a chromosomal disorder in which girls have only one \(X\) chromosome. It affects about 1 in 2000 girls. About 1 in 10 girls with Turner's syndrome suffers from narrowing of the aorta. (a) In a group of 4000 girls, what is the probability that no girls are affected with Turner's syndrome? That one girl is affected? Two? At least three? (b) In a group of 170 girls affected with Turner's syndrome, what is the probability that at least 20 of them suffer from an abnormal narrowing of the aorta?

Toss a coin with probability of heads \(0.3\) five times. Let \(X\) be the number of tails. Find (a) \(P(X=2)\). (b) \(P(X \geq 1)\).

A family has three children. Assuming a \(1: 1\) sex ratio, what is the probability that at least one child is a boy?

Assume that \(A\) and \(B\) are disjoint and that both events have positive probability. Are they independent?

Suppose that a set contains \(n\) elements. Argue that the total number of subsets of this set is \(2^{n}\).

Use the following facts: Cystic fibrosis is an inherited disorder that causes abnormally thick body secretions. About 1 in 2500 white babies in the United States has this disorder. About 3 in 100 children with cystic fibrosis develop diabetes mellitus, and about 1 in 5 females with cystic fibrosis is infertile. Find the probability that, in a group of 5000 newborn white babies in the United States, at least 4 babies suffer from cystic fibrosis.

Roll a fair die six times. Let \(X\) be the number of times you roll a 6 . Find the probability mass function.

The total maximum score on a calculus exam was 100 points. The mean score was 74 and the standard deviation was 11 . Assume that the scores are normally distributed. (a) Determine the percentage of students scoring 90 or above. (b) Determine the percentage of students scoring between 60 and 80 (inclusive). (c) Determine the minimum score of the highest \(10 \%\) of the class. (d) Determine the maximum score of the lowest \(5 \%\) of the class.