Conditional Probability in Statistics

Definition

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$ , which reads as "the probability of event $A$ given event $B$ ". The concept of conditional probability is crucial in understanding the relationship between events and is widely used in various statistical analyses.

Formula

The conditional probability of event $A$ given event $B$ is calculated using the following formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ where:

• $P(A \cap B)$ is the joint probability of events $A$ and $B$ occurring together.
• $P(B)$ is the probability of event $B$ .

Example 1: Drawing Cards from a Deck

Consider a standard deck of 52 playing cards. What is the probability of drawing an Ace given that the card drawn is a Spade?

• Event $A$ : Drawing an Ace.
• Event $B$ : Drawing a Spade.

There are 4 Aces in the deck, and there are 13 Spades in the deck, including 1 Ace of Spades. Thus:

• $P(A) = \frac{4}{52} = \frac{1}{13}$
• $P(B) = \frac{13}{52} = \frac{1}{4}$
• $P(A \cap B)$ : Probability of drawing the Ace of Spades is $\frac{1}{52}$ .

Using the conditional probability formula: $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{52}}{\frac{1}{4}} = \frac{1}{52} \times \frac{4}{1} = \frac{4}{52} = \frac{1}{13}$

So, the probability of drawing an Ace given that the card drawn is a Spade is $\frac{1}{13}$ .

Example 2: Medical Diagnosis

Suppose a certain disease affects 1% of a population. A test for the disease has the following properties:

• The test is 99% accurate if a person has the disease (true positive rate).
• The test is 95% accurate if a person does not have the disease (true negative rate).

We want to find the probability that a person has the disease given that they tested positive.

• Event $D$ : Person has the disease.
• Event $T$ : Test is positive.

Given:

• $P(D) = 0.01$ (1% of the population has the disease)
• $P(T|D) = 0.99$ (Test is positive if the person has the disease)
• $P(T|D^c) = 0.05$ (Test is positive if the person does not have the disease, false positive rate)

To find $P(D|T)$ , we need $P(T)$ , the total probability of testing positive: $P(T) = P(T|D)P(D) + P(T|D^c)P(D^c)$ $P(T) = (0.99 \times 0.01) + (0.05 \times 0.99)$ $P(T) = 0.0099 + 0.0495$ $P(T) = 0.0594$

Now, using Bayes' theorem: $P(D|T) = \frac{P(T|D)P(D)}{P(T)}$ $P(D|T) = \frac{0.99 \times 0.01}{0.0594}$ $P(D|T) = \frac{0.0099}{0.0594}$ $P(D|T) \approx 0.1667$

So, the probability that a person has the disease given that they tested positive is approximately 0.1667, or 16.67%.

Conclusion

Conditional probability is a fundamental concept in statistics that allows for the analysis of dependent events. By understanding how to calculate and interpret conditional probabilities, statisticians and researchers can gain deeper insights into the relationships between events and make more informed decisions based on data. The examples provided illustrate how conditional probability can be applied in practical scenarios, from drawing cards to medical diagnostics.