Independent and Dependent Events in Probability: Formulas, Examples, and Differences

Probability is all about measuring how likely an event is to happen. 

In some situations, the outcome of one event doesn’t have any effect on the other event. In other situations, what happens first influences the chances of what may happen next. 

These two different situations are known as independent events and dependent events. It’s important to understand this difference to calculate probability correctly.

Although the purpose of both is to calculate the likelihood of an event happening, the formulas and reasoning behind them are different.

In this article, I’ll provide a thorough guide to independent and dependent events in probability. Let’s get started!

What is an event in probability?

In probability, an event is an outcome or a group of outcomes from a random experiment. It is actually a subset of the total sample space. Let’s understand it with an example.

While rolling a die, getting a 4 is an event. Getting an even number, 2, 4, or 6, or an odd number, 1, 3, or 5, is also an event. It depends on what we are interested in measuring.

An event is usually represented by a letter like A or B. The probability is written as P(A) or P(B). For an event with equally likely outcomes (like in the rolling die example):

P(A) = Number of favorable outcomes ÷ Total number of possible outcomes

So, the probability of rolling a 4 (event) will be written as:

P(A) = 1/6

What are independent events?

These are events in which the outcome of one event doesn’t affect the chances of the next event. That’s why they are called “independent.” Their probabilities don't depend on one another.

They can occur in sequence. But the probabilities remain unchanged. That’s because independent events don’t affect the odds of each other.

Formula for independent events

The basic formula for two independent events is:

P(A and B) = P(A) × P(B)

Here, A and B are two independent events.

The multiplication rule is used here because event A does not change the probability of event B.

Examples of independent events

Let’s understand independent events with the help of these examples.

1. Tossing a coin twice

Suppose you want to calculate the chances of getting heads while tossing a coin twice. 

The probability of getting heads on the first toss is ½, and similarly it is also ½ on the second. As the first toss doesn’t affect the second toss, so:

P(heads and heads) = 1/2 × 1/2 = 1/4

2. Rolling two dice

Let’s say you want to roll a 3 on the first die and a 5 on the second.

The probability of rolling a 3 on the first die is ⅙, and the probability of rolling a 5 on the second die is also ⅙. Because the two rolls are independent:

P(3 and 5) = 1/6 × 1/6 = 1/36

What are dependent events?

These are the events in which the outcome of one event has an effect on the probability of the next event. They are in contrast to independent events.

In dependent events, the first event impacts the conditions or information relevant to the next event. That’s why the chances of the coming events change.

Formula for dependent events

The main formula for two dependent events is:

P(A and B) = P(A) × P(B | A)

Here, P(B | A) represents the probability of B occurring given that A has occurred.

For dependent events, you have to calculate the conditional probability P(B∣A). That's because the first event has a direct effect on the second.

Examples of dependent events

See the following examples to understand dependent events.

1. Drawing two balls without replacement

Suppose a bag contains a total of 8 balls. 5 are red, 3 are blue. You randomly select a ball without returning it. The probability of selecting a red ball in the first event is:

P(red first) = 5/8

If the picked ball is red, it means that there are 4 red balls remaining now and also 7 balls in total. So, the second event: 

P(red second | red first) = 4/7

The probability of getting two red balls will be:

5/8 × 4/7 = 20/56 = 5/14 

2. Choosing students randomly without replacement

Let’s say a class has 20 students. 12 are girls and 8 are boys. If one girl is selected initially and not returned to the group, it changes the probability of another girl’s selection subsequently.

For the first event:

P(girl first) = 12/20

For the second event:

P(girl second ∣ girl first) = 11/19

The probability of a girl’s random selection in both events becomes:

12/20 × 11/19 = 132/380 = 33/95

Independent vs. dependent events

The main difference between the two is whether one event changes the probability of the next event. Here’s a breakdown:

Feature	Independent Events	Dependent Events
Effect between events	One event doesn’t affect the other	One event affects the other
Second probability	Remains unchanged	Changes depending on the first event
Common example	Tossing a coin twice	Drawing two balls randomly from a bag without putting one back
Formula	P(A and B) = P(A) × P(B)	P(A and B) = P(A) × P(B | A)

A simple way to identify the difference between the two is asking yourself:

“Does the first event change the conditions for the second?”

If the answer is yes, the events are dependent. If it’s a no, the events may be independent.

How to calculate the probability of independent and dependent events

The same process can be used for both types of events. You just need to change the formula depending on whether one event affects another. Here’s a complete step by step process that you can follow for calculating the probability of both independent and dependent events.

Identify the events

First of all, clearly identify the events and understand which one is event A and which one’s event B. 

Determine if the events are independent or dependent

After that, ask whether the outcome of the first event has an impact on the outcome of the second event. If yes, the events are dependent. If not, the events are independent.

Figuring it out is crucial as it determines which formula to use later.

Find the probability of the first event

Calculate the first event’s probability P(A) using favorable outcomes and the total possible outcomes.  

This step usually stays the same for both the dependent and independent events.

Calculate the probability of the second event

For independent events, use P(B) directly, as nothing changes here since P(A) has no impact on P(B).

For dependent events, you have to calculate the conditional probability P(B | A). That’s because the first event has a direct effect on the second.

Apply the right formula

For independent events, use:

P(A and B) = P(A) × P(B)

For dependent events, go with:

P(A and B) = P(A) × P(B | A)

Simplify the result

When you have the final result for probability, you can further simplify it. You can either convert it into a decimal or percentage form (if the question requires that).

Verify the answer

The final step is to verify if the answer really makes sense.

Always remember that the probability is never greater than 1 and never lower than 0 (when converted into a decimal). And when converted to a percentage, it is always between 0% and 100%. This rule may give you a basic idea about the accuracy of your final answer.

To be completely sure about your answer, you can verify it using an online Probability Calculator.

Common mistakes in probability questions

Students may often make mistakes while solving probability questions. Some of the most common ones include:

• Choosing a formula even before understanding the situation/event

• Confusing independent and mutually exclusive events

• Updating the numerator but not the denominator

• Adding instead of multiplying

• Ignoring order in counting

A little attention before calculation can help avoid most of these errors. First, identify the events carefully. Then apply the right rule and check your final answer against the given situation to confirm that it makes sense. Focused practice along with regular revision techniques can help you catch these mistakes before the exam. 

Practice questions

Here are a few practice questions for students to solve after learning how to calculate probability:

Q1. A spinner has 4 equal sections numbered 1, 2, 3, and 4. It is spun twice. What is the probability of getting a 2 on both spins?

Q2. A box has 12 pens. 7 are green, and 5 are black. Two pens are selected one after another without replacement. What is the probability that both pens are green?

Q3. A fair die is rolled thrice. What is the probability of rolling an even number on all three rolls?

Q4. A coin is tossed twice. What’s the probability of getting heads on both occasions? 

Wrapping Up

Both independent and dependent events are used to measure probability. The main difference between the two is whether one event affects the other or not. Independent events don’t affect each other’s outcome. Whereas the first outcome has an effect on the next outcome in dependent events.

Keeping that basic concept in mind, the calculation of probability becomes easier. It’s also important to use the correct values and the right formula before finding the final answer. It’s always wise to verify your answer before finalizing it.

By practicing different questions related to probability, you can solve them more confidently and accurately with time.