Whenever you toss a fair coin, the probability of the coin landing on the head or tail always remains the same. Consider a bag containing some different coloured balls. If you pick any ball from the bag and put it back and again pick a ball, then also the probability of picking a ball of certain colour always remains the same. But if you perform the same task of picking a ball from a bag at random and do not replace the ball, does the probability of picking a certain coloured ball remains the same?

The answer is No. Such types of events where the probability of an event is affected by some other event(s) are known as dependent events. Conversely, if the probability of an event is not affected by other occurrences of other events, such events are called independent events.

Let’s understand what are independent events and their theorems and properties with examples.

## What are Independent Events?

The two events are said to be independent events if the outcome of one event does not affect the outcome of another. In other words, we can say that if one event does not influence the probability of another event, it is called an independent event.

Here are some independent events:

- You flip a coin and get a head and you flip a second coin and get a tail.
- The two coins don’t influence each other.
- The probability of rain today and the probability of my garbage being collected today.
- The garbage will be collected, rain or shine.

Here are some non-independent events:

- You draw one card from a deck and it’s black and you draw a second card and it’s black.
- By removing one black card, you made the probability of drawing a second one slightly smaller. Technically this is called ‘sampling without replacement’.
- The probability of a severe hailstorm today and the probability that the local airport will be closed to flights sometime today. Severe storms mean that airports sometimes need to close.
- The chance that you are hungry right now and the chance that you’re eating right now. Obviously one leads to the other eventually.

## Formula for Independent Events

If $\text{A}$ and $\text{B}$ are two events then the conditional probability of happening of $\text{A}$ given that $\text{B}$ has already happened is given by $\text{P}(\text{A} | \text{B}) = \frac{\text{P}(\text{A} \cap \text{B})}{\text{P}(\text{B})}$. ——————– (1)

Also, if $\text{A}$ and $\text{B}$ happen to be independent events then $\text{P}( \text{A}| \text{B}) = \text{P}(\text{A})$. ——————————– (2)

From (1) and (2), we get

$\frac{\text{P}(\text{A} \cap \text{B})}{\text{P}(\text{B})} = \text{P}(\text{A})$

$=> \text{P}(\text{A} \cap \text{B}) = \text{P}(\text{A}) . \text{P}(\text{B})$

Therefore, two events are said to be independent events if the probability of the intersection of the events is equal to the product of the probabilities of the two events.

## Difference Between Independent and Dependent Events

The following are the differences between independent and dependent events.

Independent Events | Dependent Events |

It refers to the occurrence of one event not affecting the probability of another event. | It refers to the occurrence of one event affecting the probability of another event. |

Formula for independent events is $ \text{P}(\text{A} \cap \text{B}) = \text{P}(\text{A}) . \text{P}(\text{B})$ | Formula for dependent events is $ \text{P}(\text{A} \cap \text{B}) = \text{P}(\text{A}) . \text{P}(\text{B | A})$ |

Example: These two events are independent. Picking a card from a deck of cards and again picking a card from the deck after replacing the first card. | Example: These two events are independent. Picking a card from a deck of cards and again picking a card from the deck without replacing the first card. |

## Examples on Independent and Dependent Events

**Example 1:** A die is rolled twice and a coin is tossed twice. What is the probability that the die will turn a $6$ each time and the coin will turn a tail both times?

Each time the die is rolled is an independent event. The probability of getting a $6$ is $\frac{1}{6}$.

So the probability of getting a $6$ when the die is rolled twice is $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$.

Similarly, the probability of getting a tail in two flips that follow each other are independent, therefore, their probability = $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

Also as the two events i.e. rolling the die and tossing a coin are independent, therefore, the probability of both events = $\frac{1}{36} \times \frac{1}{4} = \frac{1}{144}$.

**Example 2:** A die is rolled $4$ times. What is the probability that each throw will return a prime number?

Getting a prime number when a die is rolled is an independent event.

The prime numbers of the die are $2$, $3$, and $5$.

Therefore, the probability of getting a prime number when a die is rolled = $\frac{3}{6} = \frac{1}{2}$.

Thus, the probability of getting a prime number every time when a die is rolled $4$ times = $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16}$.

**Example 3:** The chance of a flight being delayed is $0.2$, what is the chance of no delay on a round trip?

The chance of a flight being delayed = $0.2$

Therefore, the chance of a flight not having a delay = $1 − 0.2 = 0.8$

Hence, the chance of no delay on a round trip = $0.8 \times 0.8 = 0.64$.

**Example 4:** Two cards are picked in succession from a well-shuffled deck of $52$ cards

- with replacement
- without replacement

Find the probability of drawing an ace both times.

- When two cards are picked in succession from a deck of $52$ cards with replacement, the event of drawing an ace each time are independent event.

Probability of drawing an ace from a pack of $52$ cards = $\frac{4}{52} = \frac{1}{13}$

Therefore, the probability of drawing an ace in each pick after replacement = $\frac{1}{13} \times \frac{1}{13} = \frac{1}{169}$

- When two cards are picked in succession from a deck of $52$ cards without replacement, the event of drawing an ace each time are dependent event.

Probability of drawing an ace from a pack of $52$ cards in the first attempt = $\frac{4}{52} = \frac{1}{13}$

And the probability of drawing an ace from a pack of $52$ cards in the second attempt = $\frac{3}{51}$

Therefore, the probability of drawing an ace in each pick without replacement = $\frac{1}{13} \times \frac{3}{51} = \frac{3}{663}$.

## Key Takeaways

- The probability of independent events occurring in sequence can be found by multiplying the results together.
- If the probability of one event does not affect the probability of another event, the events are independent.
- If the probability of one event affects the probability of another event, the events are dependent.

## Practice Problems

- Which of the following events is
**not**independent?- Rolling a die and getting a $6$.
- Tossing a coin and getting a head.
- Tossing two coins and getting two heads.
- Choosing a blue ball from a bag containing blue and yellow balls, not replacing it, and then choosing another blue ball from the same bag.
- A student in class $5$ has brown hair and a height of $3$ feet.

- The probability that Ravi forgets his PE kit is $0.3$. The probability that Saurabh forgets his PE kit is $0.1$. Calculate the probability that both Ravi and Saurabh forget their PE kits on the same day.
- A boy tosses three fair coins. Find the probability that all three coins land on tails.
- Jagdish has two bags of mixed sweets. He picks one sweet from each bag. The probability that Jagdish picks a lollipop from the first bag is $\frac{1}{6}$ and the probability that he picks a lollipop from both bags is $\frac{1}{15}$. What is the probability that he picks a lollipop from the second bag?
- Rohit flips a biased coin twice. The probability that Rohit gets two heads is $0.64$. Find the probability that he gets a head on a single coin flip.

## FAQs

### What do you mean by independent events?

The two events are said to be independent events if the outcome of one event does not affect the outcome of another.

### What are examples of independent events?

Examples of independent events are

a) You flip a coin and get a head and you flip a second coin and get a tail.

b) The two coins don’t influence each other.

c) You roll a die and get $3$ and you roll the die a second time and get $5$.

### How do you know if events are independent?

The two events are said to be independent events if the probability of the intersection of the events is equal to the product of the probabilities of the two events, i.e., the two events $\text{A}$ and $\text{B}$ are independent, if $\text{P}(\text{A} \cap \text{B}) = \text{P}(\text{A}) . \text{P}(\text{B})$.

### What is the difference between mutually exclusive and independent events?

The difference between mutually exclusive and independent events is: a mutually exclusive event can simply be defined as a situation when two events cannot occur at the same time whereas an independent event occurs when one event remains unaffected by the occurrence of the other event.

### What is the difference between independent and dependent events?

Independent events are events where the probability of one event does not affect the probability of a second event. Dependent events are events where the probability of one event does affect the probability of a second event.

**What are independent events give 2 examples.**

The two examples of independent events are

a) Picking a card from a well-shuffled deck of cards and picking another card after replacing the first card.

b) Getting $1$ on the first roll of a die and $5$ on the second roll of the die.

## Conclusion

The two events are said to be independent events if the outcome of one event does not affect the outcome of another. You can check whether the two events $\text{A}$ and $\text{B}$ are independent, if $\text{P}(\text{A} \cap \text{B}) = \text{P}(\text{A}) . \text{P}(\text{B})$.

## Recommended Reading

- What Are Mutually Exclusive Events – Definition, Formula & Examples
- Probability Rules(With Formulas & Examples)
- What is Experimental Probability – Definition, Approach, Formula & Examples
- What is Theoretical Probability – Definition, Approach, Formula & Examples
- What is Probability – Definition, Terminologies, Uses & Examples