When Mendel conducted his famous hybridization experiments, he used peas with green pods and yellow pods. One experiment involved crossing peas in such a way that 25% (or 145) of the 580 offspring peas were expected to have yellow pods. Instead of getting 145 peas with yellow pods, he obtained 152. Assume that Mendel’s 25% rate is correct. Is there strong evidence to suggest that Mendel’s rate of 25% is incorrect?

(a) Referring to Problem 1.69, calculate the total volume (in liters) of air an adult breathes in a day. (b) In a city with heavy traffic, the air contains 2.1 $\times 10^{-6}$ L of carbon monoxide (a poisonous gas) per liter. Calculate the average daily intake of carbon monoxide in liters by a person.

Fusing “nanofibers” with diameters of 100 to 300 nm gives junctures with very small volumes that would potentially allow the study of reactions involving only a few molecules. Estimate the volume in liters of the junction formed between two such fibers with internal diameters of 200 nm. The scale reads 1 $\mu$m.

A human brain weighs about 1 kg and contains about 10$^{11}$ cells. Assuming that each cell is completely filled with water (density = 1 g/mL), calculate the length of one side of such a cell if it were a cube. If the cells are spread out in a thin layer that is a single cell thick, what is the surface area in square meters?

In April 1986, an accident at a nuclear power plant in Chernobyl, Ukraine, scattered radioactive fallout for hundreds of miles. In assessing the biological effects of the radiation, researchers found mosses to be especially valuable as organisms for monitoring the damage. Radiation damages organisms by causing mutations. Explain why it is faster to observe the genetic effects of radiation on mosses than on other types of plants. Imagine that you are conducting tests shortly after a nuclear accident. Using potted moss plants as your experimental organisms, design an experiment to test the hypothesis that the frequency of mutations decreases with the organism’s distance from the source of radiation.

A tool-and-die machine shop produces extremely high-tolerance spindles. The spindles are 18-inch slender rods used in a variety of military equipment. A piece of equipment used in the manufacture of the spindles malfunctions on occasion and places a single gouge somewhere on the spindle. However, if the spindle can be cut so that it has 14 consecutive inches without a gouge, then the spindle can be salvaged for other purposes. Assuming that the location of the gouge along the spindle is best described by a uniform distribution, what is the probability that a defective spindle can be salvaged?

The reliability of a piece of equipment is frequently defined to be the probability p that the equipment performs its intended function successfully for a given period under specific conditions (Render and Heizer, Principles of Operations Management , 1995). Because p varies from one point in time to another, some reliability analysts treat p as if it were a random variable. Suppose an analyst characterizes the uncertainty about the reliability of a particular robotic device used in an automobile assembly line by means of the following distribution:

$$f ( p ) = \left\{ \begin{array} { l l } { 1 } & { 0 \leq p \leq 1 } \\ { 0 } & { \text { otherwise } } \end{array} \right.$$

Find the mean and variance of p .

A bus is scheduled to stop at a certain bus stop every half hour on the hour and the half hour. At the end of the day, buses still stop after every 30 minutes, but because delays often occur earlier in the day, the bus is never early and is likely to be late. The director of the bus line claims that the length of time a bus is late is uniformly distributed and the maximum time that a bus is late is 20 minutes. a. If the director’s claim is true, what is the expected number of minutes a bus will be late? b. If the director’s claim is true, what is the probability that the last bus on a given day will be more than 19 minutes late? c. If you arrive at the bus stop at the end of a day at exactly half-past the hour and must wait more than 19 minutes for the bus, what would you conclude about the director’s claim? Why?

A classical probability problem involves a king who wanted to increase the proportion of women by decreeing that after a mother gives birth to a son, she is prohibited from having any more children. The king reasons that some families will have just one boy, whereas other families will have a few girls and one boy, so the proportion of girls will be increased. Is his reasoning correct? Will the proportion of girls increase?

One student conducted the simulation described in the mentioned example and stated that the probability of getting a sequence of six 0s or six 1s is 0.977. What is wrong with that statement?

A problem that has attracted much attention in recent years is the Monty Hall problem, based on the old television game show "Let's Make a Deal," hosted by Monty Hall. Suppose you are a contestant who has selected one of three doors after being told that two of them conceal nothing, but that a new red Corvette is behind one of the three. Next, the host opens one of the doors you didn't select and shows that there is nothing behind it. He then offers you the choice of sticking with your first selection or switching to the other unopened door. Should you stick with your first choice or should you switch? Develop a simulation of this game and determine whether you should stick or switch.

Assume that you have access to a computer that can randomly generate whole numbers between any two values. Describe how this computer can be used to simulate the rolling of a pair of dice.

Develop a simulation for finding the probability that when 50 people are randomly selected, at least 3 of them have the same birth date. Describe the simulation and estimate the probability.

Listed below are randomly selected values from STATDISK and the TI-83/84 Plus calculator. Both samples are obtained by selecting a uniform distribution of whole numbers between 1 and 1000 inclusive. Are the two samples independent or dependent? Explain.