4.45 How many courses? At a small liberal arts college, students can register for one to six courses. Let X be the number of courses taken in the fall by a randomly selected student from this college. In a typical fall semester, 5% take one course, 5% take two courses, 13% take three courses, 26% take four courses, 36% take five courses, and 15% take six courses. Let X be the number of courses taken in the fall by a randomly selected student from this college. Describe the probability distribution of this random variable.

4.46 Make a graphical display. Refer to the previous exercise. Use a probability histogram to provide a graphical description of the distribution of X.

4.47 Find some probabilities. Refer to Exercise 4.45.

4.48 Use the uniform distribution. Suppose that a random variable X follows the uniform distribution described in Example 4.25 (page 240). For each of the following events, find the probability and illustrate your calculations with a sketch of the density curve similar to the ones in Figure 4.9 (page 240).

4.49 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

4.50 Use of Twitter. Suppose that the population proportion of Internet users who say that they use Twitter or another service to post updates about themselves or to see updates about others is 19%.¹² Think about selecting random samples from a population in which 19% are Twitter users.

4.51 Use of Twitter. Refer to the previous exercise. Find the probabilities for the number of Twitter users in a sample of size 2.

4.52 Households and families in government data. In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and of family size in the United States:

Make probability histograms for these two discrete distributions, using the same scales. What are the most important differences between the sizes of households and families?

4.53 Discrete or continuous. In each of the following situations, decide whether the random variable is discrete or continuous and give a reason for your answer.

4.54 Texas hold ’em. The game of Texas hold ’em starts with each player receiving two cards. Here is the probability distribution for the number of aces in two-card hands:

4.55 Tossing two dice. Some games of chance rely on tossing two dice. Each die has six faces, marked with one, two, . . . , six spots called pips. The dice used in casinos are carefully balanced so that each face is equally likely to come up. When two dice are tossed, each of the 36 possible pairs of faces is equally likely to come up. The outcome of interest to a gambler is the sum of the pips on the two up-faces. Call this random variable X.

4.56 Nonstandard dice. Nonstandard dice can produce interesting distributions of outcomes. You have two balanced, six-sided dice. One is a standard die, with faces having one, two, three, four, five, and six spots. The other die has three faces with one spot and three faces with six spots. Find the probability distribution for the total number of spots Y on the up-faces when you roll these two dice.

4.57 Spell-checking software. Spell-checking software catches “nonword errors,” which are strings of letters that are not words, as when “the” is typed as “eth.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the following distribution:

4.58 Find the probabilities. Let the random variable X be a random number with the uniform density curve in Figure 4.9 (page 240). Find the following probabilities:

4.59 Uniform numbers between 0 and 2. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be all numbers between 0 and 2. Call the random number generated Y. Then the density curve of the random variable Y has constant height between 0 and 2, and height 0 elsewhere.

4.60 The sum of two uniform random numbers. Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.12.

4.61 How many close friends? How many close friends do you have? Suppose that the number of close friends adults claim to have varies from person to person with mean μ = 9 and standard deviation σ = 2.4. An opinion poll asks this question of an SRS of 1100 adults. We will see in the next chapter that, in this situation, the sample mean response has approximately the Normal distribution with mean 9 and standard deviation 0.0724. What is , the probability that estimates μ to within ±1?

4.62 Normal approximation for a sample proportion. A sample survey contacted an SRS of 700 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We will see in the next chapter that, in this situation, the proportion of the sample who voted has approximately the Normal distribution with mean μ = 0.56 and standard deviation σ = 0.019.