Find the probability of patients who survived

Question

Upon arrival at a hospital's emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year:

• 2% of the emergency room patients were critical

• 24% of the emergency room patients were serious

• the rest of the emergency room patients were stable

• 31% of the critical patients died

• 6% of the serious patients died and

• 3% of the stable patients died.

Find the probability that the patient who survived was categorized as serious upon arrival.

Answer 1

We are trying to find out the likelihood, or probability, that a patient "who survived was categorized as serious upon arrival."

First, let's make sure we know all of the probabilities involved in this sample space. Of the patients admitted to this hospital's emergency room:

--2% or 0.02 were critical

--24% or 0.24 were serious

--"the rest" = 1 - (0.02 + 0.24) = 74% or 0.74 were stable

Ok, great! Now we know the proportions of the conditions of admitted patients--a patient was either categorized as critical, serious, or stable. This also means that a patient could not be in more than one of these categories.

Since we know now that the proportion of patients categorized as serious is 0.24, we are halfway done! Now, let's see who survived and who died.

The problem states that:

--the proportion of critical patients who died is 0.31

--the proportion of serious patients who died is 0.06

--the proportion of stable patients who died is 0.03

Adding these together, we get 0.31 + 0.06 + 0.03 = 0.4, or the proportion of all patients who died after being admitted.

This means that 0.6 is the proportion of patients who survived. Hang in there, we are almost there!

If we know a patient survived, we have three possibilities, and the problem gives us the information we need

--a critical patient survived: 1-0.31 = 0.69

--a serious patient survived: 1-0.06 = 0.94

--a stable patient survived: 1-0.03 = 0.97

We care about the serious patients for our answer, so let's find the probability of a serious patient surviving:

--probability of being categorized serious: 0.24

--probability of surviving overall: 0.6

--0.24 * 0.6 = 0.144, the chance of a randomly selected patient being both serious and also surviving. Careful, this is not our answer! We need to find out, if we know a patient survived, the probability of that survivor having been admitted as serious.

Nearly done! Since there are different rates of survival for the three different categories of patients, we need to find out the probability of a patient who survived being serious. To do that, we multiply the individual rates of survival by the probability of a patient being in each category:

--critical: 0.02 * 0.69 = 0.014

--serious: 0.24 * 0.94 = 0.226

--stable: 0.74 * 0.97 = 0.7178

Next: (0.226)/(0.014 + 0.226 + 0.7178) = 0.226/0.9578 = 0.236.

So the probability that a surviving patient was categorized as serious upon admittance is 0.236.