For this blog, let’s take a more question-oriented informal style, and hope it is as impacting as a more formal style. So, we will start by defining and clarifying aspects of probability in the context of Medicine, and then promote a deeper understanding of it by giving several simple examples.
What is Probability?
Probability (p) indicates how likely an event (a) is to occur.
So, p(a) = 0 indicates that the probability of event, a, occurring is highly unlikely, or will unlikely happen.
p(a) = 1 indicates that the probability of event, a, occurring is very likely, or better expressed as guaranteed to happen.
Probabilities can be expressed as decimals: 0 to 1, or as percentages: 0 % to 100 %.
The Question of Events
Strategically, matching a type of situation with a type of event makes problem-solving easier. Each representative type of situation has an associated solution based on the classification of the event.
It is time to go concrete.
So, we speak of the likelihood of an event occurring. What do we consider events in Medicine?
Medical events include but are not limited to:
- Infection with a specific viral disease,
- Recovery after receiving a specific treatment,
- Dextrocardia in females, and
- Colorblindness in adult males.
Independent Events versus Mutually Exclusive Events
Independent events:
The outcome of one event is unaffected by the outcome of the other event, e.g. a female with polycystic ovarian syndrome and with dextrocardia.
Non-independent events:
The outcome of one event is affected by the outcome of the other, e.g. the probability of selecting two primigravid patients from the waiting room of an OB/GYN clinic.
Mutually Exclusive Event:
The outcome of one event blocks or precludes the outcome of the other event, e.g. the probability of being Rhesus positive or being Rhesus negative. You cannot be both at the same time.
Non-mutually Exclusive Event:
The outcome of one event is unaffected by the outcome of the other event, e.g. a 35-year-old male with obesity and metabolic syndrome, or being male and having blood type B.
Calculations
Some of our calculations examples require knowledge of the ABO blood group system. Following, we have included a brief summary just in case you might have forgotten some aspects of this system, so that you will be able to make sense of the calculations that require knowledge of it.
Facts needed for calculations about blood groups:
- Everyone can receive blood from blood group O.
- Only persons with blood group A or AB can receive blood from blood group A.
- Only persons with blood group B or AB can receive blood from blood group B, and
- Only persons with blood group AB can donate blood to blood group AB.
Warmup calculation 1:
For the blood groups, we have O, A and AB are 47%, 42% and 3%, respectively, and B is unknown for the UK.
If we randomly select anyone in the UK, what is the probability of that person having blood group B?
Here, we use the Rule: the SUM of all probabilities must add up to 1 or 100 %.
Solution:
p(B) = 1 – [p(O) + p(A) + p(AB)]
- = 1 – [0.47 + 0.42 + 0.03]
- = 1 – 0.92
- = 0.08 or 8% (This is the answer).
Warmup calculation 2:
For the blood groups, we have O, A, B and AB are 47%, 42%, 8% and 3%, respectively, and B is no longer unknown.
If we randomly select anyone in the UK, what is the probability of that person NOT having blood group B?
Here, we use the rule: the SUM of all probabilities must add up to 1 or 100 %.
Solution:
- p(NOT B) = 1 – p(B)
- Since p(B) is 0.08
- p(NOT B) = 1 – 0.08.
Therefore, p(NOT B) = 0.92 (This is the answer).
Final Warmup calculation:
For the blood groups, we have O, A, B and AB are 47%, 42%, 8% and 3%, respectively.
If we randomly select anyone in the UK, what is the probability that this person cannot donate blood to everyone?
We use the rule: the SUM of all probabilities must add up to 1 or 100 %.
Solution:
p(NOT O) = 1 – p(O)
- = 1 – 0.47
and therefore,
- P(NOT O) = 0.53 (This is the answer).
So, now we are all warmed up, let’s look at a few more worked examples.
Independent Events Calculation
Person with polycystic ovarian syndrome, a, and with dextrocardia, b.
Rule: for Independent Events, probabilities are multiplied.
p(a AND b) = p(a)p(b).
Independent Calculation 1:
Data:
County X has a population of 10000.
- 4500 are males and 5500 are females,
- 550 have polycystic ovarian syndrome, and
- 2 has dextrocardia.
Question:
From a certain fictitious county, County X, if we randomly select anyone, what is the probability that this person has polycystic ovarian syndrome and dextrocardia?
Equation: p(a AND b) = p(a)p(b)
Solution:
p(a AND b) = p(a) p(b)
- = (550/10000)(2/10000)
- = (0.055)(0.0002).
- = 0.000011 (This is the answer).
Non-Independent Events Calculation
The probability of selecting two primigravid patients from the waiting room of an O B G Y N clinic.
Non-independent Events: probabilities are multiplied
p(a AND b) = p(a)p(a with one less).
Non-Independent Calculation 2:
Data:
County X has a population of 10000.
- 4500 are males and 5500 are females,
- 550 have polycystic ovarian syndrome,
- 2 have dextrocardia,
- 200 are in the OB/GYN waiting room, and
- 60 are primigravid.
Question:
From a certain fictitious county, if we randomly select anyone, what is the probability of selecting two primigravid patients from the waiting room of an OB/GYN clinic?
Equation: p(a AND a-1) = p(a)p(a-1)
Solution:
p(a AND b) = p(a)p(a-1)
- = (60/200)(59/199)
- = (0.30)(0.296482412)
- = 0.088944724 = 0.09 or 9% (This is the answer).
Mutually Exclusive Events Calculation
The probability of being Rhesus D positive or being Rhesus D negative.
For Mutually Exclusive Events: probabilities are added, and their sum should be 1.
p (a, or b) = p(a) + p(b).
Mutually Exclusive Events Calculation 3:
Data:
County X has a population of 10000.
- 4500 are males and 5500 are females.
- 9800 are Rhesus D positive and 200 are Rhesus D negative.
Question:
From a certain fictitious county, County X, if we randomly select anyone, what is the probability of that person being Rhesus D positive or being Rhesus D negative?
Equation: p (a, or b) = p(a) + p(b)
Solution:
p (a, or b) = p(a) + p(b)
- = (9800/10000) + (200/10000)
- = (9800 + 200)/(10000)
- = 10000/10000 = 1 or 100% (This is the answer).
Non-Mutually Exclusive Events Calculation
The probability of a person at random being male or having blood type B.
For Non-mutually Exclusive Events: probabilities are added, and their products are then subtracted.
p (a, or b) = p(a) + p(b) – p(a)p(b).
Non-Mutually Exclusive Events Calculation 4:
Data:
County X has a population of 10000.
4500 are males and 5500 are females.
If the blood groups have the same distribution as that of the UK, we have O, A, B and AB are 47%, 42%, 8% and 3%.
Question:
From a certain fictitious county, County X, if we randomly select anyone, what is the probability of that person being male or having blood type B?
Equation: p (a, or b) = p(a) + p(b) – p(a)p(b)
Solution:
p (a, or b) = p(a) + p(b) – p(a)p(b)
- = (4500/10000) + (800/10000) – (4500/10000)(800/10000)
- = (4500 + 800)/(10000) – (4500 X 800)/(10000 X 10000)
- = 5300/10000 – 3600000/100000000.
- = 0.530 – 0.036
- = 0.494 or 49% (This is the answer).
We hope that we helped you understand probabilities a bit more, or at least encouraged you to read up more on probabilities as well as practice more problems.
If you would like to see any inclusion or exclusion to this blog, please let us know.
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