Dependent Events

Dependent events are events in probability where the outcome of one event affects or changes the probability of another event. In other words, once one event occurs, it affects the likelihood of another event happening afterward.
 

For example, if you have a jar containing five red marbles and five blue marbles. You pick one marble and do not put it back. Then you pick another marble.

If the first marble is red, there are now fewer red marbles left in the jar.

Because the first pick affects the second pick, these are dependent events.

In probability, an event is any possible outcome or a group of outcomes from a random experiment. It represents what we observe or measure during the experiment. Events are classified by how their outcomes relate to each other. The main types of events are: 
 

• Simple Events
   
• Compound Events
   
• Mutually Exclusive Events
   
• Dependent Event
   
• Independent Event
  
   

Independent and dependent events differ in whether the outcome of one event influences the probability of the other—the difference between independent and dependent events. 

Dependent events are events in which the outcome of one event influences or changes the result of another. These events play an important role in probability, decision-making, and real-world predictions. Below are the key properties of dependent events:
 

• In dependent events, the outcome of one event affects the likelihood of the next. Once an event occurs, the information gained from it changes the conditions for the following event.
   
• The probability of dependent events changes after each event. So, after the first event changes the situation, the probabilities for the next event are different.The probability of dependent events changes after each event. So, after the first event changes the situation, the probabilities for the next event are different.
   

• When one event occurs, it can alter the number of remaining possible outcomes. For example, drawing a card from a deck without replacing it reduces the total number of cards for the next draw.
   

• Dependent events do not always have a direct or obvious connection. For example, bad weather might increase the number of car accidents. Weather and accidents are different, but they affect each other.
   

• The sequence in which events happen can affect the probabilities of dependent events. For example, drawing cards one after another without replacement is different from drawing with replacement.

The probability of dependent events is based on conditional probability, which tells us how likely one event is to occur given that another has already happened. In dependent events, the outcome of the first event changes the probability of the second event.
 

For example, a box has three red balls and two blue balls. You pick one ball and do not put it back. Then you pick a second ball. Since the first ball is not replaced, the number of balls changes. This means the probability of the second pick depends on what was picked first, so the events are dependent.

The probability of dependent events is the chance that one event occurs after another. The formula for finding the probability of dependent events uses conditional probability. It is written as:
 

\(P(B | A) = {{P (A ∩ B) \over P (A)}}\)
 

Where P(A ∩B) is the probability that both events A and B occur
P(A) is the probability that event A happens.

To find the probability of a dependent event, we use the concept of conditional probability. This helps us determine how likely an event is to occur after another event has already occurred.
 

If event A happens first and event B occurs next, the probability of event B after A is written as P(B | A). The formula is: 
 

\(P(B | A) = {P (A ∩ B) \over P (A)}\)

For example, a jar contains three red candies and two blue candies. You pick one candy and do not put it back. Then you pick another candy. If you like a red candy first, what is the probability that you will enjoy a blue candy second?

To find the probability of picking blue candy, we use the formula: 

 \(P(B | A) = {P (A ∩ B) \over P (A)}\)

Let, the probability of picking a red candy be event A and the probability of picking a blue candy be B. 

Finding P(A):

There are 3 red and 2 blue candies, so the total is 5
 

P(A) = \(3\over 5\)

After picking a red, the total candies left = 4

So, the remaining blue candies = 2

So, \(P(B|A) = {{2\over 4}} = {{1\over 2}}\)

So, the probability of picking a blue candy second after picking a red candy first is \(1 \over 2\).

Helping children understand dependent events becomes much easier when learning is grounded in simple examples and hands-on activities. Here are a few tips and tricks to master dependent events. 

• Memorize the key formula for a dependent event: \(P(B | A) = {{P (A ∩ B)) \over (P (A) }} \).
   
•  Adjust the sample size after the first Event, since the total number of outcomes changes. 
   
• Parents can ask the child to explain how one Event affects the next. This helps them slow down, think clearly, and avoid mistakes in probability calculations.
   
• Teachers can use visual models like diagrams, probability trees, or colored counters to explain how outcomes change after each Event. 
   
• Check whether the second Event depends on the first Event. If the result of Event B changes after Event A happens, then the events are dependent.

We use the concept of dependent events on a daily basis. It is widely used in environmental sciences and various other fields.

•  Weather Prediction: Meteorologists use dependent events to accurately predict the weather conditions and to find the probability that the weather tomorrow depends on today's weather conditions. 

• Sports Strategies: Sports analysts use dependent events to predict whether the team is going to win or lose based on their previous performance and their score in the current game.

• Education: Educational institutes use dependent events to determine whether a student is going to pass based on the students’ previous performance in earlier exams.

• Finance: dependent events are used to assess credit risk, where the likelihood of loan default depends on prior payment history.
   
• Medical Diagnosis: Doctors use dependent events when diagnosing patients — for example, the probability of having a certain disease may depend on family history, lifestyle, or previous test results.
   
• Marketing and Customer Behavior: Businesses analyze dependent events to predict future purchases. A customer’s likelihood of buying a product depends on their previous buying history or interaction with similar items.