Introduction to Basics of Probability #2

We got introduced to probability in Part 1. Now let us continue with some more basics of probability.
In this post we will learn about some additional terms.

Independence

When doing a probability study, we look at events and the chances of their occurrence. For Example we have Event A and B. If the probability of event A is not affected by the probability of event B, and vice versa, then these two events can be said to be independent of each other.
The independence of two events can be tested using a comparison of the event probability and the conditional probability. The formula for independence test is as follows.

independence - basics of probability

Medical Screening

Screening or Medical Screening is a practical application of Probability that is used to screen for a likelihood of an event. The Reason it is commonly called as Medical screening is because of its global usage in the medical industry in testing for various diseases. Testing for diseases such as Malaria, HIV, Corona Virus Pandemic of 2020 and many more, follow this procedure. The testing probability is always arranged in the form of a box given below.

The above table shows the values of random test results performed on a group randomly selected people.

Sensitivity

Sensitivity is the probability that a test gives Positive output provided a test subject actually has the disease. The Sensitivity of the sample test given above, is given by P(Positive|Affected) = 95/101 = 0.9405 or 94.05%
If the result is positive but the person is not affected then this is known commonly as a FALSE POSITIVE.
False positive Probability is given by P(Positive|Not Affected) = 5/99 = 0.0505 or 5.05%

NOTE: These 2 conditional probabilities DO NOT add to 1.

Specificity

Specificity is the probability that a test gives Negative output provided a test subject does not have the disease. The Specificity of the sample test given above, is given by P(Negative|Not Affected) = 94/99 = 0.9494 or 94.94%
If the result is negative but the person is affected then this is known commonly as a FALSE NEGATIVE.
False Negative Probability is given by P(Negative|Affected) = 6/101 = 0.0594 or 5.94%

NOTE: These 2 conditional probabilities DO NOT add to 1.

Events

Events are the possible occurrences or outcomes of an experiment or observation. we have already discussed independent events, dependent or Conditional Events. Events can also be classified into another type of category.

Inclusive Events

Inclusive Events are events that may occur simultaneously or independently. When designing a probability experiment The statistician must ensure that he does not have any inclusive events in the probability, as that will render the experiment a failure in most cases. Inclusive events may or may not be independent of each other. An example would be the chance that a ball is rolling or bouncing. Logically speaking, if a ball is rolling, then it is not bouncing. But when a ball is bouncing, it may be rolling.

Mutually Exclusive Events

Mutually exclusive events are those that events that cannot occur at the same time. Continuing the above example. in different terms. Observing if a ball is stationary, rolling or bouncing. Out of these 3 events, The event marked as stationary is mutually exclusive from the other two events. That is, the ball cannot be stationary while rolling or bouncing, and vice versa.

Complementary Events

Complementary events are events that mutually exclusive events that cover all possibilities of events possible. This usually means that there are only 2 possible outcomes. The ball is moving or it is not. It is raining, or it is not. Such experiments or observations have only two possible outcomes and can be calculated by the formula

P(A) = 1- P(B)

In the example of Medical screening Sensitivity and FALSE NEGATIVE are complementary events. FALSE POSITIVE and Specificity are the second pair of Complementary Events.

For more examples on Event types click here.

These are the Basics of Probability and will help build the foundation of statistics and Statistical studies.

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