Chapter 12

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 José, Hilary, Peter, and Jessica sit on a bench if Peter and Jessica want to be next to each other?

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 1000 children with cystic fibrosis, at least 25 will develop diabetes mellitus.

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 goats on a farm is \(123 \mathrm{lb}\), and the standard deviation is \(9 \mathrm{lb}\). If the weights are normally distributed, determine what percentage of goats weigh (a) between 110 and \(130 \mathrm{lb}\), (b) less than \(100 \mathrm{lb}\), and \((\mathbf{c})\) more than \(150 \mathrm{lb}\).

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 $$ P(A \cup B)=1-P\left(A^{c}\right) P\left(B^{c}\right) $$

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

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 250 women with cystic fibrosis, no more than 60 are infertile.

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 ?

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 \(X\).

Suppose that you pick a number \(X\) at random from the interval \((0, a)\). If \(P(X \geq 1)=0.2\), find \(a\).

Cystic Fibrosis 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 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 hall 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?

An urn contains four red, seven green, and two white balls. You draw a ball at random, note its color, and replace it. You repeat these steps four times. Let \(X\) denote the number of red balls and \(Y\) the number of green balls. Find \(P(X+Y=2)\).

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 many 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?

Suppose a woman has a hemophilic brother and one healthy son. Suppose furthermore that neither her mother nor her father were hemophilic but that her mother was a carrier for hemophilia. Find the probability that she is a carrier of the hemophilia gene.

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 all of the patients be assigned?

Blood Test To test for a disease that has a prevalence of \(1 \mathrm{in}\) 100 in a population, blood samples of 10 individuals are mixed and the mixed blood is then tested. What is the probability that the test result is negative (i.e., 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 study in which patients are divided into four groups of 25 patients each. In how many ways can this be done if each patient is assigned to exactly 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 \((x+y)^{4}\).

Toss a fair coin 10 times. Let \(X\) denote the number of heads. What is the probability that \(X\) is within one standard deviation of its mean?

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 \(X=\min \left(X_{1}, X_{2}, \ldots,\right.\), \(\left.X_{n}\right)\) (a) Compute \(P(X>x)\). (b) Show that \(P(X>x / n) \rightarrow e^{-x}\) as \(n \rightarrow \infty\).

Four cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of at least one ace?

In how many ways can four red and five black cards be selected from a standard deck of cards if cards are drawn without replacement?

A true-false exam has 20 questions. Find the expected number of correct answers if a student guesses the answers at random.