Independent and Dependent Events
In probability, events do not always behave the same way. Sometimes the occurrence of one event has no effect on another event. Other times, one event directly affects the outcome of another.
To handle such situations correctly, we classify events as independent or dependent.
Independent Events
Two events are called independent if the occurrence of one event does not affect the probability of the other event.
In simple terms, what happens first does not change what happens next.
Numerical Example
A fair coin is tossed and a die is rolled.
- Probability of getting Heads = 1/2
- Probability of getting a 3 on the die = 1/6
The coin toss does not affect the die roll.
Probability of getting Heads and 3:
1/2 × 1/6 = 1/12
Real-World Example
Weather conditions today and the result of a dice roll are independent events. One does not influence the other.
Dependent Events
Two events are called dependent if the occurrence of one event affects the probability of the other event.
This usually happens when items are selected without replacement.
Numerical Example
A bag contains 3 red balls and 2 blue balls. Two balls are drawn one after another without replacement.
Probability of drawing a red ball first:
3/5
After one red ball is removed, the bag contains 2 red and 2 blue balls.
Probability of drawing a red ball second:
2/4
Probability of drawing two red balls:
3/5 × 2/4 = 6/20 = 3/10
Real-World Example
Selecting students from a class for a team without replacement is an example of dependent events, because each selection changes the remaining group.
Key Difference Between Independent and Dependent Events
| Aspect | Independent Events | Dependent Events |
|---|---|---|
| Effect of first event | No effect on second event | Affects second event |
| Probability changes? | No | Yes |
| Common scenario | Coin toss, dice roll | Card draw without replacement |
When Events Become Independent Again
If items are selected with replacement, the events become independent because the original conditions are restored.
For example, if a card is drawn and then put back into the deck before drawing again, the probabilities do not change.
Quick Check
Are the following events independent or dependent?
Drawing two cards from a deck without replacement.
Practice Quiz
Question 1:
Tossing two coins at the same time is an example of which type of events?
Question 2:
Why are events without replacement usually dependent?
Question 3:
Which rule is commonly used to calculate probabilities for independent events?
Mini Practice
A jar contains 4 white balls and 6 black balls. Two balls are drawn one after another.
- Are the events independent or dependent if the balls are not replaced?
- Would they become independent if the first ball is replaced?
What’s Next
In the next lesson, we will study Conditional Probability, which explains how probability changes when additional information is known.