We are interested in the probability of getting at least two boys in a three-child family. The assumptions are that the probability of a boy is 0.5 (i.e., 50%) and that the sexes of the children are independent. We will estimate this probability using the five-step simulation method described below.
The five-step method carries out a simulation to experimentally estimate theoretical probabilities. As in the text, the required probability will be estimated by carrying out the five steps.
The five steps for this problem are:
First, we need to define the appropriate box model, which is simply a box containing balls having numbers written on them. This step is very important because it specifies the probability model, or population, for whatever statistical problem is being studied. The most important step is to decide the characteristics of the balls to be put in the box: how many balls, what value is written on each, and how many of each kind of ball. See Step 1 of the program (Define the box model) for the entry screen.
The Value column contains the values that are to be written on the balls, and the Count column specifies the number of balls with each value. We need a probability model in which boys and girls are selected with equal probability. Thus, for the current problem, we need two values: 0 (a girl) and 1 (a boy), each with a count of 1. Since the counts are equal, girls (0 values) and boys (1 values) are selected with equal probability. By default the counts are 1, so fill in the values in the first two positions of the Value column. Click on Show Boxes to see the balls, their values, and the number of times each ball occurs. Click on Histogram to see a visual representation of the ball values and the number of times each occurs. Click on the Next button to go to Step 2.
A trial consists of drawing balls from the box with replacement until we have simulated a 3-child family. To define the required process, click on the Define the sample: pop-up menu and select the Draw n with replacement menu item. Specify 3 draws in the n= field. Click on the Next button to go to Step 3.
We are interested in the number of boys in a 3-child family. Click on the Statistic of interest (X) menu and select the Sum menu item. This will cause the program to sum the selected values in a trial, which consists of drawing three balls (n = 3), each with a 0 or 1. The sum will simply count the number of boys since each ball has either a 1 (for a boy) or a 0 (for a girl). We must specify the event X >= 2 boys in a 3-child family. From the Event of interest (if required) menu select the X >= a menu item, where in this case X represents the sum statistic. Fill in 2 in the a= field. Your selections in this step actually specify P(sum >= 2) = P(#boys >= 2), i.e., the probability of getting 2 or more boys in a 3-child family. Click on the Next button to go to Step 4.
The six buttons labeled 1, 10, 100, 1000, 5000, 10,000 determine the number of times one can repeat the trial. It is possible to click on the buttons repeatedly. For example, clicking on 1000 and then 5000 results in 6,000 trials. Click on the 1 button. The computer will then draw 3 balls from the box. The actual draws are given in the Current outcome field. For example, your trial may look like this: 0 1 0.
Thus, on this trial, the first draw was a girl (0), the next was a boy (1), and third was a girl (0). The numerical outcome (statistic of interest) is the sum, 1 in this case, which appears in the area labeled Sum. Your trial's outcome may be different. The sum of the numbers (outcomes) is the number of boys. If the sum is 2 or 3, then we have a Success as defined by the probability of interest. If the sum is less than 2 (i.e., 0 or 1), then we have a Failure. Using this terminology, we want to estimate the probability of a success.
One trial is not enough, since the estimate will be either a 0 or a 1. To perform more trials, just click on the 1 button nine more times, each time noticing what happens. At this point, you may have gotten at least 1 of each of the possible outcomes (0, 1, 2, and 3), but we cannot tell reliably how likely it is to get a 2 or 3. What is the empirical estimate of the event probability? We need far more runs to get a good estimate of the empirical probability of the event Sum >= 2. Click the 100 button to give a total of 110 runs. The program will run these trials, each time showing the trial outcome (quickly) at the Current simulation: field and writing the numerical outcome in the Sum area. What is estimate of the event probability now?
The total number of trials so far has been 110, which is given in the # of Simulations field. Click the 1000 button to get a total of 1110 runs. What is estimate of the event probability now? Finally, click on the 5000 button and examine the event probability. The estimate of the event probability should be converging towards a value (the theoretical probability)? Guess the value of the theoretical probability.
Click on the Next button to go to Step 5.
We now want to examine the summary statistics from our simulation consisting of 6,110 trials. The Event of Interest probability is the focus of the problem. Your estimate (Expt Prob) should be close to 0.5, the theoretical probability of getting at least 2 boys in a 3-child family. When we study the binomial distribution, we will see how this number is computed.
Besides the experimental probability, you will see other summary statistics in Step 5. Specifically, the mean and standard deviation are important. (The median will always be 1 or 2. Why?) The mean should be close to 1.5, the theoretical mean, i.e., on the average a family has 1.5 boys.
A frequency histogram of the 6,110 outcomes is also provided. Note that the distribution is approximately symmetric and takes on the values 0, 1, 2, and 3. Also, getting 1 boy or getting 2 boys in a 3-child family is about three times as likely as getting no boys or 3 boys. These probabilities will be computed exactly when we study the binomial distribution.Once you have navigated to a new step, you can always go back to a previous step by clicking on one of the numbers (1, ..., 5) in the Step tool palette. For example, go back to Step 4 and run another 1000 trials. Note that the # of Simulations increases each time more simulations are run. Why? You can click Reset in Step 4 to start new trials.