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).

a. If 1 of the 159,511 patients in the study is randomly selected, what is the probability that the subject was transported by helicopter?

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

Forty-seven thousand six hundred thirty-seven patients were transported by helicopter while 111874 were transported by ground.

The probability of any event is computed as the ratio of counts of outcomes that favor the event to the total counts of outcomes.

In case there are multiple events, the multiplication rule of probability is applicable. The two possible cases are:

• Without replacement: The probability of the successive events is dependent and changes with each draw.
• With replacement: The probability for the event remains constant throughout all draws.

It is tedious to calculate the probability of occurrence of multiple events simultaneously if selections are made without replacement. To avoid lengthy calculations, the following rule is applied.

Whenthe sample size is equal to or less than 5% of the population size,all events can be considered independent (condition of ‘with replacement’ applies) . Thus, the probability of a number of events to occur together is computed by simply multiplying the individual probabilities of all the events.

a.

The total number of subjects is equal to 159511.

The number of subjects transported by helicopter is equal to 47637.

The number of subjects transported by ground is equal to 111874.

The probability of selecting a subject who was transported by helicopter is given as follows:

$$\begin{gathered} P\left(A\right)=\frac{47637}{159511} \\ =0.299 \end{gathered}$$

Therefore, the probability of selecting a subject who was transported by helicopter is equal to 0.299.

b.

Let B be the event of selecting five subjects that were transported by helicopter without replacement.

The sample size is 5.

The population size is 159511.

$\frac{5}{159511}\times 100=0.0031\%$

As the sample size is less than 5% of the population size, the events can be considered independent.

The probability of selecting five subjects that were transported by helicopter without replacement is given as follows:

$$\begin{gathered} P\left(B\right)=P\left(A\right)\times P\left(A\right)\times P\left(A\right)\times P\left(A\right)\times P\left(A\right) \\ ={\left(P\left(A\right)\right)}^5 \\ ={\left(0.299\right)}^5 \\ =0.00238 \end{gathered}$$

Therefore, the probability of selecting five subjects transported by helicopter without replacement is equal to 0.00238.