1160 LearnersLast updated on June 5th, 2025
Independent Events
Independent events are the events that occur together, but the outcome of one event does not affect the outcome of the other in any manner. In this topic, we are going to talk about independent events and how they are different from dependent events.
What Are Independent Events?
Two events are said to be independent events if the result of one event does not impact the result of another event. In math, we would say that the outcome of one event does not impact the probability of another event, this is what we call independent events. One such example would be winning the lottery and going on a picnic. Both events do not influence the other, making this an independent event.
We usually show the probability of independent events with the formula:
P(A ∩ B) = P(A) × P(B)
If two events A and B are independent, then the probability of both events happening is equal to A ∩ (and) B
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Difference Between Independent Events and Dependent Events
Probability has two types of events: Independent events and dependent events. Here are a few differences between the two events:
| Independent Events | Dependent Events |
| When two events do not impact the probability of each other's outcome, we call it independent events. | The occurrence of one event affects the probability of another event. |
| Example: Watching a movie on your laptop and booking a cab. | Example: Drawing twp cards from a deck without replacement |
| Formula: P(A and B) = P(A) × P(B) |
Formula: P(A and B) = P(A) × P(B after A) |
What are the rules for Independent Events?
When learning about independent events there are a few certain rules that must be followed such as:
Rule of multiplication:
We use this rule to find the probability of any events that occur simultaneously. The rule of multiplication is expressed as P(A ∩ B) = P(A) × P(B)
Rule of addition:
The rule of addition allows us to determine the probability if at least one of the events occurs. It is expressed as (A ∪ B) = P(A) + P(B) - P(A ∩ B)
How to find The Probability of Independent Events?
To find the probability of independent events, we use a formula:
P(A and B) = P(A) × P(B)
Where:
P(A and B) is the probability that both events occur
P(A) is the probability of event A.
P(B) is the probability of event B
If these statements are true, then A and B are independent events.
Step 1: Calculate the individual probabilities so first, we find the probability of P(A) and then the probability of P(B)
Step 2: Then we calculate the joint probability, which is P(A and B)
Step 3: Determine whether it is independent or not by using the formula P(A and B) = P(A) × P(B). If it is not true, then events are dependent.
What are the methods to find Independent Events?
We know how to find the probability of independent events, but we need to first identify whether an event is independent or not:
- Check if both events can occur in any order. If it does, then we proceed to the next step.
- Now we try to determine whether the outcomes affect the other event. If it does not, then the event is considered independent. If an event does impact the outcome of the other, then it is considered dependent.
Real-life Applications on Independent Events
There are many uses of independent events. Let us now see the uses and applications of independent events in different fields:
Coin Tossing in Sports:
We use independent events in sports where in football, the coin toss at the start of the game is independent of the match’s outcome. Each toss has a 50% chance of landing heads or tails, no matter what happened before.
Rolling Dice in Board Games:
We use the concept of independent events in games like Monopoly or Yahtzee, where rolling a die is independent of the previous rolls. The probability of rolling 6 remains 1/6 on each throw.
Lottery Draws and Gambling:
We use the concept of independent events in lottery draws and gambling, where each lottery ticket has an equal chance of winning, independent of the previous draws. In a roulette game, the ball landing on red is independent of past spins.
Common Mistakes and How to Avoid Them in Independent Events
When understanding the concept of independent events, students tend to make mistakes. Here, are some common mistakes and their solutions
Mistake 1
Confusing Independent and Dependent Events
Students should always remember to check if one event impacts the probability of the other. If removing an item changes the total count, then the event is dependent.
Mistake 2
Incorrectly Applying the Multiplication Rule
Students should remember the multiplication rule on independent events. Always check the independence before multiplying the probabilities.
Mistake 3
Not Considering Real-World Context
Students should consider the external factors before treating the events as independent events.
For example, assuming that two students passing an exam are independent when they may have studied together. So, they must consider the real-world contexts before assuming a given event is independent.
Mistake 4
Overlooking Sample Space Changes
Students should remember that if an event changes the total number of outcomes, the events are not independent
Mistake 5
Assuming that Correlation Implies Independence
Students should know that correlation does not mean independence. They should always check if one event affects the probability of the other.
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Solved examples on Independent Events
Problem 1
What is the probability of getting heads on the first toss and tails on the second toss?
0.25
Explanation
P(Heads) = 0.5
P(Tails) = 0.5
We use the formula: P(A and B) = P(A) × P(B)
P(Heads and Tails) = 0.5 × 0.5 = 0.25
The probability of getting heads on the first toss and tails on the second toss is 0.25
Problem 2
What is the probability of getting tails on a coin toss and a 4 on a die roll
1/12 or 0.0833 (approximately)
Explanation
P(Tails) = ½ = 0.5
P(4 on a die) = 1/6 ≈ 0.1667
We use the formula: P(A and B) = P(A) × P(B)
P(Tails and 4 on a die) = 1/2 × 1/6 = 1/12
Problem 3
What is the probability of drawing an Ace from a standard 52-card deck and rolling a 6 on a die?
1/78
Explanation
P(Ace) = 4/52 = 1/13
P(6 on a die) = 1/6
We use the formula: P(A and B) = P(A) × P(B)
P(Ace and 6 on a die) = 1/13 × 1/6 = 1/78
Problem 4
What is the probability of rolling a 2 on the first roll and a 4 on the second roll with a die?
1/36
Explanation
P(2 on a die) = 1/6
P(4 on a die) = 1/6
We use the formula: P(A and B) = P(A) × P(B)
P(2 on a die and 4 on a die) = 1/6 × 1/6 = 1/36
Problem 5
What is the probability of rolling an even number and drawing a red card from a standard deck (there are 26 red cards out of 52)?
0.25
Explanation
P(even number on die) = 3/6 = 0.5
P(Red card) = 26/52 = 0.5
We use the formula: P(A and B) = P(A) × P(B)
P (even number on die and red card) = 0.5 × 0.5 = 0.25.
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FAQs on Independent Events
1.How do we check if two events are independent?
2.Are mutually exclusive events also independent events?
3.How do you calculate the probability of two independent events occurring together?
4.If two events are independent, does it mean they are equally likely?
5.If there are two independent events, can they have a probability of 1?
6.How can children in United States use numbers in everyday life to understand Independent Events?
7.What are some fun ways kids in United States can practice Independent Events with numbers?
8.What role do numbers and Independent Events play in helping children in United States develop problem-solving skills?
9.How can families in United States create number-rich environments to improve Independent Events skills?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
