In Exercises 29 and 30, find the probabilities and indicate when the “5% guideline for cumbersome calculations” is used.




Medical Helicopters In a study of helicopter usage and patient survival, results were obtained from 47,637 patients transported by helicopter and 111,874 patients transported by ground (based on data from “Association Between Helicopter vs Ground Emergency Medical Services and Survival for Adults with Major Trauma,” by Galvagno et al., Journal of the American Medical Association, Vol. 307, No. 15).


b. If 5 of the subjects in the study are randomly selected without replacement, what is the probability that all of them were transported by helicopter?

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that all 5 randomly selected subjects were transported by helicopter, given that the selection is made without replacement. This involves calculating probabilities from a finite population.
2. Step 2: Identify the total population and the subset of interest. The total population consists of 47,637 patients transported by helicopter and 111,874 patients transported by ground, making the total population size 47,637 + 111,874 = 159,511. The subset of interest is the 47,637 patients transported by helicopter.
3. Step 3: Use the formula for probability without replacement. The probability of selecting 5 helicopter patients consecutively without replacement is calculated as the product of conditional probabilities: \( P = \frac{47,637}{159,511} \times \frac{47,636}{159,510} \times \frac{47,635}{159,509} \times \frac{47,634}{159,508} \times \frac{47,633}{159,507} \).
4. Step 4: Apply the '5% guideline for cumbersome calculations' if applicable. The guideline states that if the sample size is less than 5% of the population, we can treat the selections as independent (i.e., with replacement) to simplify calculations. Here, the sample size (5) is much smaller than 5% of the population (159,511), so the guideline can be applied.
5. Step 5: If applying the guideline, approximate the probability using the binomial distribution. Treat the probability of selecting a helicopter patient as \( p = \frac{47,637}{159,511} \), and calculate \( P = p^5 \). This simplifies the calculation significantly compared to the exact method.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chance that all selected subjects were transported by helicopter. Understanding basic probability principles, such as the multiplication rule for independent events and the concept of combinations, is essential for solving the problem.

Sampling Without Replacement

Sampling without replacement means that once an item is selected from a population, it cannot be selected again. This affects the probabilities of subsequent selections, as the total number of items decreases with each selection. In the given question, this concept is crucial because the probability of selecting all helicopter patients changes as each patient is chosen.

5% Guideline for Cumbersome Calculations

The 5% guideline is a rule of thumb used in statistics to simplify calculations when dealing with large populations. It suggests that if the sample size is less than 5% of the population, the probabilities can be approximated as if sampling were done with replacement. This guideline can help determine whether the complexity of calculations can be reduced in the context of the helicopter and ground transport study.