Chapter 12

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

Suppose that the number of seeds a plant produces is normally distributed, with mean 142 and standard deviation \(31 .\) Find the probability that in a sample of five plants, at least one produces more than 200 seeds. Assume that the plants are independent.

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 \(2^{n}\).

Turner's syndrome is a rare chromosomal disorder in which girls have only one \(X\) chromosome. The condition affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome suffers from an abnormal 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?

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.

A family has four children. Assuming a \(1: 1\) sex ratio, what is the probability that no more than two children are girls?

Assume that the probability that an insect species lives more than five days is \(0.1\). Find the probability that, in a sample of size 10 of this species, at least one insect will still be alive after five days.

In how many ways can Brian, Hilary, Peter, and Melissa sit on a bench if Peter and Melissa want to be next to each other?

A loaded die has probability \(0.5\) of rolling a 6 and probability \(0.1\) of rolling each of the other five numbers. Find the probability of rolling a 6 three times in a row.

The mean weight of female students at a small college is \(123 \mathrm{lb}\), and the standard deviation is \(9 \mathrm{lb}\). If the weights are normally distributed, determine what percentage of female students weigh (a) between 110 and \(130 \mathrm{lb}\), (b) less than \(100 \mathrm{lb}\), and (c) more than \(150 \mathrm{lb}\).

In Problems \(36-37\), we discuss the inheritance of red-green color blindness. Color blindness is an X-linked inherited disease. \(A\) woman who carries the color blindness gene on one of her \(X\) chromosomes, but not on the other, has normal vision. A man who carries the gene on his only \(X\) chromosome is color blind. If a woman with normal vision who carries the color blindness gene on one of her \(X\) chromosomes has a child with a man who has normal vision, what is the probability that their child will be color blind?

(a) Use a Venn diagram to show that $$ (A \cup B)^{c}=A^{c} \cap B^{c} $$ (b) Use your result in (a) to show that if \(A\) and \(B\) are independent, then \(A^{c}\) and \(B^{c}\) are independent. (c) Use your result in (b) to show that if \(A\) and \(B\) are independent, then $$ P(A \cup B)=1-P\left(A^{c}\right) P\left(B^{c}\right) $$ \(12.3 .4\)

Paula, Cindy, Gloria, and Jenny have dinner at a round table. In how many ways can they sit around the table if Cindy wants to sit to the left of Paula?

A loaded die is weighted so that rolling a 4 is three times as likely as rolling any of the other numbers. You roll the die twice and record the sum of the two numbers. What is the probability that the sum is equal to 7 .

Suppose that you pick a number at random from the interval \((0,4)\). What is the probability that the first digit after the decimal point is a 3 ?

In Problems \(36-37\), we discuss the inheritance of red-green color blindness. Color blindness is an X-linked inherited disease. \(A\) woman who carries the color blindness gene on one of her \(X\) chromosomes, but not on the other, has normal vision. A man who carries the gene on his only \(X\) chromosome is color blind. If a woman with normal vision who carries the color blindness gene on one of her \(X\) chromosomes has a child with a man who is red-green color blind, what is the probability that their child has normal vision?

A screening test for a disease shows a positive result in \(95 \%\) of all cases when the disease is actually present and in \(10 \%\) of all cases when it is not. If the prevalence of the disease is 1 in 50 and an individual tests positive, what is the probability that the individual actually has the disease?

In how many ways can you form a committee of three people from a group of seven if two of the people do not want to serve together?

An urn contains four green and six blue balls. You draw a ball at random, note its color, and replace it. You repeat these steps four times. Let \(X\) denote the total number of green balls you obtain. Find the probability mass function of \(X\).

Cystic fibrosis is an autosomal recessive disease, which means that two copies of the gene must be mutated for a person to be affected. Assume that two unaffected parents who each carry a single copy of the mutated gene have a child. What is the probability that the child is affected?

A screening test for a disease shows a positive result in \(95 \%\) of all cases when the disease is actually present and in \(10 \%\) of all cases when it is not. If a result is positive, the test is repeated. Assume that the second test is independent of the first test. If the prevalence of the disease is 1 in 50 and an individual tests positive twice, what is the probability that the individual actually has the disease?

In how many ways can you form two committees of three people each from a group of nine if (a) no person is allowed to serve on more than one committee? (b) people can serve on both committees simultaneously?

An urn contains three blue and two white balls. You draw a ball at random, note its color, and replace it. You repeat these steps three times. Let \(X\) denote the total number of white balls. Find \(P(X \leq 1)\).

An urn contains three red and two blue balls. You remove two balls without replacement. What is the probability that the two balls are of a different color?

A bag contains two coins, one fair and the other with two heads. You pick one coin at random and flip it. What is the probability that you picked the fair coin given that the outcome of the toss was heads?

A collection contains seeds for four different annual and three different perennial plants. You plan a garden bed with three different plants, and you want to include at least one perennial. How many different selections can you make?

Suppose that you pick five numbers at random from the interval \((0,1)\). Assume that the numbers are independent. What is the probability that all numbers are greater than \(0.7 ?\)

An urn contains five blue and three green balls. You remove three balls from the urn without replacement. What is the probability that at least two out of the three balls are green?

You pick 2 cards from a standard deck of 52 cards. Find the probability that the first card was a spade given that the second card was a spade.

In diploid organisms, chromosomes appear in pairs in the nuclei of all cells except gametes (sperm or ovum). Gametes are formed during meiosis, a process in which the number of chromosomes in the nucleus is halved; that is, only one member of each pair of chromosomes ends up in a gamete. Humans have 23 pairs of chromosomes. How may kinds of gametes can a human produce?

Assume that \(20 \%\) of all plants in a field are infested with aphids. Suppose that you pick 20 plants at random. What is the probability that none of them carried aphids?

Suppose that \(X_{1}, X_{2}\), and \(X_{3}\) are independent and uniformly distributed over \((0,1)\). Define $$ Y=\max \left(X_{1}, X_{2}, X_{3}\right) $$ Find \(E(Y) .\) [Hint: Compute \(P(Y \leq y)\), and use it to deduce the density of \(Y .]\)

You select 2 cards without replacement from a standard deck of 52 cards. What is the probability that both cards are spades?

Sixty patients are enrolled in a small clinical trial to test the efficacy of a new drug against a placebo and the currently used drug. The patients are divided into 3 groups of 20 each. Each group is assigned one of the three treatments. In how many ways can the patients be assigned?

To test for a disease that has a prevalence of 1 in 100 in a population, blood samples of 10 individuals are pooled and the pooled blood is then tested. What is the probability that the test result is negative (the disease is not present in the pooled blood sample)?

Suppose that \(X_{1}, X_{2}\), and \(X_{3}\) are independent and uniformly distributed over \((0,1)\). Define $$ Y=\min \left(X_{1}, X_{2}, X_{3}\right) $$ Find \(E(Y) .\) [ Hint: Compute \(P(Y>y)\), and use it to deduce the density of \(Y .]\)

You select 5 cards without replacement from a standard deck of 52 cards. What is the probability that you get four aces?

One hundred patients wish to enroll in a small study in which patients are divided into four groups of 25 patients each. In how many ways can this be done if no patient is to be assigned to more than one group?

Suppose that a box contains 10 apples. The probability that any one apple is spoiled is 0.1. (Assume that spoilage of the apples is an independent phenomenon.) (a) Find the expected number of spoiled apples per box. (b) A shipment contains 10 boxes of apples. Find the expected number of boxes that contain no spoiled apples.

Suppose that you wish to simulate a random experiment that consists of tossing a coin with probability \(0.6\) of heads 10 times. The computer generates the following 10 random variables: \(0.1905\), \(0.4285,0.9963,0.1666,0.2223,0.6885,0.0489,0.3567,0.0719\), \(0.8661\). Find the corresponding sequence of heads and tails.

An urn contains four green, six blue, and two red balls. You take three balls out of the urn without replacement. What is the probability that all three balls are of different colors?

Expand \((x+y)^{4}\).

Suppose that you wish to simulate a random experiment that consists of rolling a fair die. The computer generates the following 10 random variables: \(0.7198,0.2759,0.4108,0.7780,0.2149,0.0348\), \(0.5673,0.0014,0.3249,0.6630 .\) Describe how you would find the corresponding sequence of numbers on the die, and find them.

An urn contains three green, five blue, and four red balls. You take three balls out of the urn without replacement. What is the probability that all three balls are of the same color?

Expand \((2 x-3 y)^{5}\)

A multiple-choice exam contains 50 questions. Each question has four choices. Find the expected number of correct answers if a student guesses the answers at random.

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are independent random variables with uniform distribution on \((0,1)\). Define \(X=\min \left(X_{1}, X_{2}, \ldots,\right.\), \(X_{n}\) ). (a) Compute \(P(X>x)\). (b) Show that \(P(X>x / n) \rightarrow e^{-x}\) as \(n \rightarrow \infty\).