Conditional Probability

Conditional probability is one of the commonly applied concepts in probability theory. In conditional probability, an event (A) is likely to occur given that another event (B) is occurring. Conditional probability can be applied when a student decides not to go on an excursion if it rains (5% probability), making the probability of going on the excursion 95%. In this example, the conditional probability is 5%. 

Conditional probability can be mathematically expressed as A given

B or as P(A|B). 

P(A|B)  = P(A∩ B)/ P(B)

Here:

P(A|B):  The probability of occurrence of A is dependent on the likelihood of B occurring. 

P(A∩ B): Probability of both events A and B occurring at the same time.

P (B): Probability of the occurrence of event B.

The three types of probability are conditional, joint, and marginal. Here, we will discuss a few key points that differentiate them:

Conditional probability:

In conditional probability, the probability of an event relies on another event in the past.
Here, Event A is drawing a four, and Event B is drawing a red card
If the probability of a card being drawn is red, the probability of drawing a four is:

P(A|B) = P(A∩ B)/ P(B) = 1/26 ÷1/2 = 2/26 = 1/13

Joint Probability:

A joint probability is the probability of two or more events occurring simultaneously. We assume that Event A is drawing a four (four of hearts or four of diamonds)  and Event B is drawing a red card. Here, we determine the probability of drawing a card that is both red and a four, meaning both conditions need to be satisfied. It can be mathematically expressed as:

P(A∩ B) = 2/25 = 1/26

Marginal Probability

The marginal probability is the probability of a single event occurring, independent of any other event.

P (A) or P(B) = ∑ P(A∩ B)

For example: 70% like reading (A)
                        50% like writing (B)
                        30% like both reading and writing (A ∩ B)
 The marginal probability of liking writing is:
P(B) = P (A ∩ B) + P (B ∩ A^C)
 

To find the conditional probability, we use the following steps:

1. Identify the events:

• The event whose probability needs to be determined (A).
   
• An event that has already occurred (B).

2. Calculate the joint probability P (A ∩ B) 

• The probability of both events occurring at the same time.

3. Determine the probability of the stated condition P(B).

4. Now, we use the formula : P(A|B) =  P (A ∩ B)  P(B)

Conditional Probability for Independent Events 

The probability of independent events calculates the probability of events whose occurrences do not depend on each other. Here, the conditional probability of two independent events, A and B, can be written as:

P(A|B) =  P(A)
P(B|A) = P(B)

These events satisfy the multiplication rule: P(A∩B) = P(A) × P(B)

Conditional Probability for Mutually Exclusive Events 

The events that do not occur at the same time are referred to as mutually exclusive. For mutually exclusive events, the conditional probability will always equal zero.
P(A|B) = 0
P(B|A) = 0

Conditional probability plays a vital role in the prediction of various real-life situations based on conditions. Whether predicting the weather or solving complex real-life problems, conditional probability has numerous uses. Let’s look at a few examples:

• Conditional probability is widely used in weather forecasting by predicting the probability of rain based on the condition that the sky is cloudy.

• In healthcare, the probability of confirming a disease if the test is positive.

• In traffic control, the probability of occurrence of accidents if the roads are overcrowded.

• Students use conditional probability to predict the chance of failing an exam under certain conditions, such as a strict evaluation.

• We use it in predicting the stock market returns. If the stocks are down, the probability of gaining returns will be low.