SECTION 4.1 EXERCISES

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For Exercise 4.1, see page 218.

Question 4.2

4.2 Are these phenomena random? Identify each of the following phenomena as random or not. Give reasons for your answers.

  1. (a) The outside temperature in your town at noon on Groundhog Day, February 2.

  2. (b) The first digit in your student identification number.

  3. (c) You draw an ace from a well-shuffled deck of 52 cards.

Question 4.3

4.3 Interpret the probabilities. Refer to the previous exercise. In each case, interpret the term probability for the phenomena that are random. For those that are not random, explain why the term probability does not apply.

Question 4.4

4.4 Are the trials independent? For each of the following situations, identify the trials as independent or not. Explain your answers.

  1. (a) You record the outside temperature in your town at noon on Groundhog Day, February 2, each year for the next five years.

  2. (b) The number of tweets that you receive on the next 10 Mondays.

  3. (c) Your grades in the five courses that you are taking this semester.

Question 4.5

4.5 Winning at craps. The game of craps starts with a “come-out” roll, in which the shooter rolls a pair of dice. If the total of the “spots” on the up-faces is 7 or 11, the shooter wins immediately (there are ways that the shooter can win on later rolls if other numbers are rolled on the come-out roll). Roll a pair of dice 25 times and estimate the probability that the shooter wins immediately on the come-out roll. For a pair of perfectly made dice, the probability is 0.2222.

Question 4.6

image 4.6 Use the Probability applet. The idea of probability is that the proportion of heads in many tosses of a balanced coin eventually gets close to 0.5. But does the actual count of heads get close to one-half the number of tosses? Let’s find out. Set the “Probability of Heads” in the Probability applet to 0.5 and the number of tosses to 50. You can extend the number of tosses by clicking “Toss” again to get 50 more. Don’t click “Reset” during this exercise.

  1. (a) After 50 tosses, what is the proportion of heads? What is the count of heads? What is the difference between the count of heads and 25 (one-half the number of tosses)?

  2. (b) Keep going to 200 tosses. Again record the proportion and count of heads and the difference between the count and 100 (half the number of tosses).

  3. (c) Keep going. Stop at 300 tosses and again at 400 tosses to record the same facts. Although it may take a long time, the laws of probability say that the proportion of heads will always get close to 0.5 and also that the difference between the count of heads and half the number of tosses will always grow without limit.

Question 4.7

image 4.7 A question about dice. Here is a question that a French gambler asked the mathematicians Fermat and Pascal at the very beginning of probability theory: what is the probability of getting at least one 6 in rolling four dice? The Law of Large Numbers applet allows you to roll several dice and watch the outcomes. (Ignore the title of the applet for now.) Because simulation—just like real random phenomena—often takes very many trials to estimate a probability accurately, let’s simplify the question: is this probability clearly greater than 0.5, clearly less than 0.5, or quite close to 0.5? Use the applet to roll four dice until you can confidently answer this question. You will have to set “Rolls” to 1 so that you have time to look at the four up-faces. Keep clicking “Roll dice” to roll again and again. How many times did you roll four dice? What percent of your rolls produced at least one 6?