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

Medical Helicopters In the same study cited in the preceding exercise, among the 47,637 patients transported by helicopter, 188 of them left the treatment center against medical advice, and the other 47,449 did not leave against medical advice. If 40 of the subjects transported by helicopter are randomly selected without replacement, what is the probability that none of them left the treatment center against medical advice?

The probability of any event is:

$P\left(A\right)=\frac{\text{Number}\text{of}\text{outcomes}\text{favorable}\text{to}\text{A}}{\text{Total}\text{outcomes}}$

The multiplication rule of probability is applied when multiple events occur simultaneously, with two possibilities.

• Without replacement: The probability of any successive event is dependent on the previous ones.
• With replacement: The probability of any successive event is independent of the previous ones.

When selections are made without replacement, it is important for every successive probability to take into account the occurrence of the previous events. To avoid complex calculations after every event, the following rule is used.

Whenthe sample size is equal to or less than 5% of the population size, all events can be considered as independent and the probability of all events to occur together is computed by simply multiplying the individual probabilities of all of them.

The total number of subjects who were transported by the helicopter is equal to 47637.

Out of these, the number of subjects who left the treatment without medical advice is equal to 188.

The number of subjects who did not leave the treatment without medical advice is equal to 47449.

Let A be the event of selecting 40 subjects who did not leave the treatment without medical advice without replacement.

The sample size is 40.

The population size is 47637.

$\frac{40}{47637}\times 100=0.0840\%$

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

The probability of selecting 40 subjects who did not leave the treatment without medical advice is given as follows:

$$\begin{gathered} P\left(A\right)=\frac{47449}{47637}\times \frac{47449}{47637}\times ...\times \frac{47449}{47637}\left(40\text{times}\right) \\ ={\left(\frac{47449}{47637}\right)}^{40} \\ ={\left(0.996\right)}^{40} \\ =0.854 \end{gathered}$$

Therefore, the probability of selecting 40 subjects who did not leave the treatment without medical advice and without replacement is equal to 0.854.