Random digits. As discussed in Chapter 3, generation of random numbers is one approach for obtaining a simple random sample (SRS). If we were to look at the random generation of digits, the mechanism should give each digit probability 0.1.

Consider the digit “0” in particular.

(a) The table of random digits (Table B) was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the first 200 digits in the table are 0s? This proportion is an estimate, based on 200 repetitions, of the true probability, which in this case is known to be 0.1.

(b) Now try software:

• Excel users: Input the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in column A. Now choose “Sampling” from the Data Analysis menu box (refer to the Appendix of Chapter 1 for installing the Data Analysis ToolPak). Enter the cell range of the digits in the Input Range box. Choose the “Random” option and input 10,000 in the Number of Samples box. Finally, select an output range in the worksheet.

• Minitab users: Do the following pull-down sequence: Calc→

Random Data→Integer. Enter “10000” in the Number of rows of data to generate box, type “c1” in the Store in column(s) box, enter “0” in the Minimum value box, and enter “9” in the Maximum box. Click OK to find 10,000 realizations of random digits outputted in the worksheet.

Whether you used Excel or Minitab, sort the 10,000 random digits from smallest to largest to make your counting easier. What proportion of the 10,000 digits in the column are 0s? Is this proportion close to 0.1?

4.6 How many tosses to get a head? When we toss a penny, experience shows that the probability (long-term proportion) of a head is close to 1/2. Suppose now that we toss the penny repeatedly until we get a head. What is the probability that the first head comes up in an odd number of tosses (1, 3, 5, and so on)?

To find out, repeat this experiment 50 times, and keep a record of the number of tosses needed to get a head on each of your 50 trials.

(a) From your experiment, estimate the probability of a head on the first toss. What value should we expect this probability to have?

(b) Use your results to estimate the probability that the first head appears on an odd-numbered toss.