Probability theory and statistical signficance: coincidence, FDA type I and II errors

A. At what point does the possibility that an event is more than coincidence become obvious? Consider this in terms of probability theory and statistical significance.

We can use the probability theory and statistical significance to know when the event is more than coincidental. For example, suppose the probability of dialysis working and saving the life of the patient is 0.7, and the probability of dialysis not able to save the life of the patient is 0.3 (this 0.4 may be due any other reason). So the probability of one death is 0.3. The probability of two out of two deaths is 0.3*0.3=.09
Now the probability of three out of three deaths is .3^3=.027
Now the probability of four out of four deaths is .3^4=.0081

If we set the significant level at 5% then as soon as we see three deaths it raises the alarm that we need to check that the event is not coincidental (the probability falling below 5%). However if we are working with significance level of 1%, then as soon as we see the probability falling below 1% we say that the events are not coincidental.

Thus, with the use of probability theory and statistical significance we can set the cutoff limits to see when the event is coincidental and when it is not.

Question B:
Type I error is rejecting a null hypothesis when it is true and Type II error is accepting a null hypothesis when it is false. In case of medical devices as the stakes are very high (death), the null hypothesis is that the error in medical equipment is more than the prescribed limit and alternate hypothesis is that the error less than prescribed limit. So if we set the significance level high, we are making more type I error. Since stakes are high, the cost of Type I error is quite high and one should reduce this error as much as one can. So generally we set significance level of 1% or .1% for medical equipments. On the other hand by setting the low Type I error, the Type II error increases that is concluding that equipment has high error when it really does not have that. Thus, we are rejecting the machines which are OK and thereby reducing the profitability. The trade off lies in the way in which the profits can be maximized for the equipment manufacturer. The cost of type I error is law suits for error equipments, compensation for deaths, loss of trust in the equipments by the doctors, etc.

The cost of type II error is loss of profitability on the equipment. Since the cost of lawsuits and compensation are very high, the type I error is set at very low level 1% or .1%.