Important Rules in Probability

1. Addition Rule:

P(A∪B) = P(A) + P(B) − P(A∩B)

The addition rule is used to find the probability of event A or event B occurring, considering the possibility of overlap. Let us understand this with a problem.

In a clinical study, two side effects, nausea (event A) and headache (event B), are observed after administering a drug. The probability of experiencing nausea is P(A) = 0.25, and the probability of experiencing a headache is P(B) = 0.30 The probability of experiencing both side effects is P(A∩B) = 0.10.

What is the probability that a patient will experience either nausea or a headache?

Using the Addition Rule:

P(A∪B) = P(A) + P(B) − P(A∩B)

Substitute the given values:

P(A∪B) = 0.25 + 0.30 − 0.10=0.45

Thus, the probability that a patient will experience either nausea or a headache is 0.45 (or 45%).

2. Multiplication Rule:

For independent events: P(A∩B) = P(A) × P(B)
For dependent events: P(A∩B )= P(A) × P(B∣A)

The multiplication rule is used to find the probability of both events A and B occurring, depending on whether the events are independent or dependent. Let us understand this with a problem.

In a healthcare setting, the probability of a patient having diabetes (event A) is P(A) = 0.15. The probability of a patient having high blood pressure (event B) is P(B) = 0.20. Assume the events are independent.
What is the probability that a patient will have both diabetes and high blood pressure?

Using the Multiplication Rule for independent events:

P(A∩B) = P(A) × P(B);
P(A∩B) = 0.15 × 0.20 = 0.03

Thus, the probability that a patient will have both diabetes and high blood pressure is 0.03 (or 3%).

Now, consider a different scenario where the probability of having high blood pressure given that a patient already has diabetes is P(B∣A) = 0.40. What is the probability that a patient has both diabetes and high blood pressure?

Using the Multiplication Rule for dependent events:

P(A∩B) = P(A) × P(B∣A) = 0.15 × 0.40 = 0.06

Thus, the probability that a patient has both diabetes and high blood pressure in the dependent scenario is 0.06 (or 6%).

Please note: In probability problems, deciding whether events are independent or dependent is crucial for applying the correct rule:

Identifying Independence:

1. Independent Events: Two events A and B are independent if the occurrence of one does not affect the occurrence of the other. For example:
Tossing a coin & rolling a die are independent since outcome of one does not influence the other.
In healthcare, consider two diagnostic tests for different diseases. If having one disease doesn’t affect the likelihood of the other, the tests’ results can be considered independent.

2. Dependent Events: Events are dependent when the outcome of one affects the probability of the other. For example:
Testing positive for a disease depends on whether the person has the disease, making the test result dependent on the actual health condition.
In healthcare, if a patient’s diagnosis of a second disease depends on a previous diagnosis, those events are dependent.

When solving problems, context clues guide you toward using either the multiplication rule for independent events or the rule for dependent events (i.e., conditional probabilities).

3. Total Probability Rule:

P(A) = P(A∩B) + P(A∩B^c)

The total probability rule helps us compute the probability of an event based on partitioning the sample space.

In a clinical trial, a new treatment for a disease is being tested. Patients are divided into two groups: those who respond to the treatment (event B) and those who do not respond (event B^c). The probability of a patient responding to the treatment is P(B)=0.6, and the probability of not responding is P(B^c)=0.4.

If the probability of recovery given a response is P(A∣B) = 0.8 and the probability of recovery given no response is P(A∣B^c) = 0.3, what is the total probability of recovery?

Using the Total Probability Rule:

P(A) = P(A∩B) + P(A∩B^c)

This can also be written as:

P(A) = P(A∣B) × P(B) + P(A∣B^c) × P(B^c);
P(A) = (0.8×0.6) + (0.3×0.4) = 0.48 + 0.12 = 0.60

Thus, the total probability of recovery is 0.60 or 60%.