Estimating Probability of at Least One Success

Make “At Least One” the Event You Count

In Estimating a Probability from Simulation Results, you learned to count trials in which an event occurs and divide by the total number of simulated trials. Here, the event is that at least one of four children is a girl. The phrase “at least one” includes one, two, three, or four girls. A trial with no girls does not count as a success.

To simulate the situation, use a model in which each child is equally likely to be a girl or a boy, and the outcomes for different children are independent. A random digit from 0 through 9 can represent one child: assign 0, 1, 2, 3, and 4 to girl, and 5, 6, 7, 8, and 9 to boy. Each outcome receives five of the ten equally likely digits, matching the model’s equal chances.

Generate four digits for each trial, keeping them together as one group. For example, 8271 represents four children: 8 is a boy, 2 is a girl, 7 is a boy, and 1 is a girl. This trial has two girls, so it is a success for the event “at least one girl.” In contrast, 9867 represents four boys and is not a success.

Notice that the event is about a group of four, not about a randomly selected child. The success count is the number of groups with at least one girl. It is not the total number of girls across all groups.

Set Up and Tally the Four-Child Trials

The digit assignment makes each child’s simulated outcome straightforward to read. For each group, count how many digits are from 0 through 4. Then mark “Yes” if that count is at least one and “No” if the count is zero. Counting the girls can help you classify a trial, but the final tally records whether the event occurred, not how many girls occurred.

As in Setting Up a Simulation Model and Simulating a Fixed Number of Trials, a clear trial definition keeps the simulation aligned with the question. Do not stop a group when the first girl appears: each trial must contain four children, even if a girl appears in the first position.

Work Through a Simulation Results Table

The following 20 trials each contain four digits. The same digit assignment is used throughout. “Girls in trial” is a check on the classification; the event of interest is whether this count is one or more.

Check the classifications by looking at the digits, not by relying on a rough visual impression of each row. For instance, trial 11 is 4026: digits 4, 0, and 2 represent girls, while 6 represents a boy, so the count is three girls and the trial is a success. Trial 18 is 9274: digits 2 and 4 represent girls, so the count is two and the trial is also a success. A trial with more than one girl still contributes only one success to the event tally.

Count “At Least One” by Counting Its Opposite

There are two consistent ways to tally this event. You can mark every trial with one or more girls directly. Or you can count the trials with zero girls—all four digits represent boys—and subtract that count from the total number of trials. This second method works because every trial either has at least one girl or has no girls; the two categories do not overlap and together include all trials.

The shortcut can be useful when zero-girl trials are easier to identify than all the different successful outcomes. It does not change the event or the denominator. It only gives another way to find the number of successes.

A frequency table reports how many trials had each outcome. The frequencies are counts, not probabilities. Add the frequencies for the qualifying outcomes to find the success count, or subtract the frequency for zero girls from the total. Both approaches give the same numerator when applied correctly.

Check the Tally in Batches

When results come in batches, record the number of no-girl trials in each batch or the number of successful trials in each batch. Keep track of how many trials each batch contains. The batch counts should add to the full number of trials; this helps catch a missed row or a trial counted twice.

Common Mistakes and AP Exam Tips

• Counting children instead of trials. In the first table there are 30 simulated girls among 80 children, but the target event is defined for groups of four. Its estimate is \(13/20\), not \(30/80\). The numerator must count successful groups.
• Counting only trials with exactly one girl. “At least one” includes every count from one through four. A trial with three girls is a success, just like a trial with one girl.
• Giving a trial multiple successes. The row 3308 contains three girls, but it is one successful trial, not three. Each group contributes either one success or zero successes to this event tally.
• Using the wrong denominator. Divide by the number of four-child trials, not the number of digits, children, or successful trials. If there are 20 groups, the denominator is 20.
• Stopping a trial after a girl appears. A trial is always four children. Keeping all four outcomes makes each simulated trial match the stated situation.
• Mixing up the digit assignment. With this model, digits 0–4 mean girl and 5–9 mean boy. Apply the same assignment consistently to every position and every trial.

A full-credit explanation identifies the event, describes what counts as one trial, gives the success count and total, and states the estimate in context. For example: “A success is a simulated four-child group with at least one girl. Thirteen of the 20 trials were successes, so the estimated probability that at least one of four children is a girl is \(13/20=0.65\), or 65%.” This makes clear that the estimate comes from simulated groups and that the numerator counts groups, not individual girls.