Independent Vs Mutually Exclusive Events (3 Key Concepts)


Independent or mutually exclusive events are important concepts in probability theory.  They help us to find the connections between events and to calculate probabilities.

So, what is the difference between independent and mutually exclusive events?  Events A and B are independent if the probability of event B is the same whether A occurs or not, and the probability of event A is the same whether B occurs or not. Events A and B are mutually exclusive if they cannot occur at the same time. Events cannot be both independent and mutually exclusive.

We often use flipping coins, rolling dice, or choosing cards to learn about probability and independent or mutually exclusive events.

In this article, we’ll talk about the differences between independent and mutually exclusive events.  We’ll also look at some examples to make the concepts clear.

Let’s get started.

Independent Vs Mutually Exclusive Events

Independent events and mutually exclusive events are different concepts in probability theory.  Two events A and B can be independent, mutually exclusive, neither, or both.

The table below summarizes the differences between these two concepts.

Independent
Events
Mutually
Exclusive
Events
P(AnB)=P(A)P(B)P(AnB)=0
P(A|B)=P(A)P(A|B)=0
P(B|A)=P(B)P(B|A)=0
P(A) does not
depend on
whether B
occurs or not
If B occurs,
A cannot
also occur.
P(B) does not
depend on
whether A
occurs or not
If A occurs,
B cannot
also occur.

What Are Independent Events?

Two events A and B are independent if the occurrence of one does not affect the occurrence of the other.  That is, the probability of event B is the same whether event A occurs or not.

Two events that are not independent are called dependent events.

What Is The Multiplication Rule For Independent Events?

You can tell that two events A and B are independent if the following equation is true:

  • P(AnB) = P(A)P(B)

where P(AnB) is the probability of A and B occurring at the same time.  This is called the multiplication rule for independent events.

We can also express the idea of independent events using conditional probabilities.  If A and B are independent events, then:

  • P(A|B) = P(A)

and

  • P(B|A) = P(B)

Let’s look at some examples of events that are independent (and also events that are not independent).

Example 1: Two Independent Events

Let’s say you have a quarter and a nickel.  Both are coins with two sides: heads and tails.

quarter coin flip
A fair coin has a 50% chance of landing on heads and a 50% chance of landing on tails when flipped.

We are going to flip both coins, but first, let’s define the following events:

  • A is the event that we flip heads on the quarter.
  • B is the event that we flip tails on the nickel.

There are two ways to tell that these events are independent: one is by logic, and one is by using a table and probabilities.

Logically, when we flip the quarter, the result will have no effect on the outcome of the nickel flip.  We can also build a table to show us these events are independent.

The table below shows the possible outcomes for the coin flips:

Coin
Flip
Outcomes
Heads
On
Nickel
Tails
On
Nickel
Heads
On
Quarter
(1/2)*(1/2)
=1/4
(1/2)*(1/2)
=1/4
Tails
On
Quarter
(1/2)*(1/2)
=1/4
(1/2)*(1/2)
=1/4
This table shows the possible outcomes
and probabilities for two coin flips
(a quarter and a nickel).

If the coins are fair, then:

  • The probability of event A (flipping heads on the quarter) is ½ or 0.5.
  • The probability of event B (flipping tails on the nickel) is ½ or 0.5.

Since all four outcomes in the table are equally likely, then the probability of A and B occurring at the same time is ¼ or 0.25.

This means that P(AnB) = P(A)P(B), since 0.25 = 0.5*0.5.

Example 2: Two Events That Are Not Independent

Let’s say you are interested in what will happen with the weather tomorrow.

Let’s define the following events:

  • A is the event that the temperature is below 32 degrees Fahrenheit all day.
  • B is the event that it snows.

These two events are not independent, since the occurrence of one affects the occurrence of the other:

  • If A does not happen, we are likely to get rain instead of snow (due to temperatures above freezing).
  • If A does happen, we are likely to get snow instead of rain (due to temperatures below freezing all day).
snow
Temperature and type of precipitation are not independent, since freezing cold air means a chance of snow instead of rain.

What Are Mutually Exclusive Events?

Two events A and B are mutually exclusive (disjoint) if they cannot both occur at the same time.  That is, event A can occur, or event B can occur, or possibly neither one – but they cannot both occur at the same time.

You can tell that two events are mutually exclusive if the following equation is true:

  • P(AnB) = 0

Simply stated, this means that the probability of events A and B both happening at the same time is zero.

Example 1: Two Mutually Exclusive Events

Let’s say you have a quarter, which has two sides: heads and tails.

We are going to flip the coin, but first, let’s define the following events:

  • A is the event that we flip heads on the quarter.
  • B is the event that we flip tails on the quarter.

These events are mutually exclusive, since we cannot flip both heads and tails on the coin at the same time.

Example 2: Two Events That Are Not Mutually Exclusive

Let’s say you have a quarter and a nickel, which both have two sides: heads and tails.

We are going to flip the coins, but first, let’s define the following events:

  • A is the event that we flip heads on the quarter.
  • B is the event that we flip heads on the quarter and tails on the dime.

These events are not mutually exclusive, since both can occur at the same time.  Specifically, if event B occurs (heads on quarter, tails on dime), then event A automatically occurs (heads on quarter).

We can also tell that these events are not mutually exclusive by using probabilities.  The probability that both A and B occur at the same time is:

  • P(AnB)
  • =P(flip heads on quarter and tails on dime)
  • =1/4

Since P(AnB) is not zero, the events A and B are not mutually exclusive.

Example 3: Two Events That Are Not Mutually Exclusive

Let’s say you are interested in what will happen with the weather tomorrow.

Let’s define the following events:

  • A is the event that the temperature is below 32 degrees Fahrenheit all day.
  • B is the event that it snows.

These two events are not mutually exclusive, since the both can occur at the same time: we can get snow and temperatures below 32 degrees Fahrenheit all day.

How To Find Probability Of 2 Independent Events

To find the probability of 2 independent events A and B occurring at the same time, we multiply the probabilities of each event together.  Remember the equation from earlier:

  • P(AnB) = P(A)P(B)

Example: Finding The Probability Of 2 Independent Events

Let’s say that you are flipping a fair coin and rolling a fair 6-sided die.  Let’s define these events:

  • A is the event that you flip heads on the coin.
  • B is the event that you roll 6 on the die.

These events are independent, since the coin flip does not affect the die roll, and the die roll does not affect the coin flip.

We can calculate the probability as follows:

  • P(AnB)
  • =P(A)P(B)  [by definition of independent events]
  • =(1/2)P(B)  [probability of flipping heads on a fair coin is 1/2]
  • =(1/2)(1/6)  [probability of rolling 6 on a fair 6-sided die is 1/6]
  • =1/12

How To Find Probability Of 3 Independent Events

To find the probability of 3 independent events A, B, and C all occurring at the same time, we multiply the probabilities of each event together.  Remember the equation from earlier:

  • P(AnB) = P(A)P(B)

We can extend this to three events as follows:

  • P(AnBnC)
  • =P((AnB)nC)
  • =P(AnB)P(C)  [since AnB and C are independent events]
  • =P(A)P(B)P(C)  [since A and B are independent events]

So, P(AnBnC) = P(A)P(B)P(C), as long as the events A, B, and C are all mutually independent, which means:

  • the occurrence of A does not affect B or C
  • the occurrence of B does not affect A or C
  • the occurrence of C does not affect A or B

Example: Finding The Probability Of 3 Independent Events

Let’s say that you are flipping a fair coin, rolling a fair 6-sided die, and rolling a fair 10-sided die.  Let’s define these events:

  • A is the event that you flip heads on the coin.
  • B is the event that you roll 6 on the 6-sided die.
  • C is the event that you roll 10 on the 10-sided die.
10 sided die
A fair 10-sided die has a 10% chance to show each of the digits 0 through 9.

These events are independent, since the coin flip does not affect either die roll, and each die roll does not affect the coin flip or the other die roll.

We can calculate the probability as follows:

  • P(AnBnC)
  • =P(A)P(B)P(C)  [by definition of independent events]
  • =(1/2)P(B)P(C)  [probability of flipping heads on a fair coin is 1/2]
  • =(1/2)(1/6)P(C)  [probability of rolling 6 on a fair 6-sided die is 1/6]
  • =(1/2)(1/6)(1/10)  [probability of rolling 10 on a fair 10-sided die is 1/10]
  • =1/120

Can Independent Events Be Mutually Exclusive?

Independent events cannot be mutually exclusive events.  Remember that if events A and B are mutually exclusive, then the occurrence of A affects the occurrence of B:

  • If A happens, then B cannot happen.
  • If B happens, then A cannot happen.

Thus, two mutually exclusive events are not independent.

***Note: if two events A and B were independent and mutually exclusive, then we would get the following equations:

  • P(AnB) = P(A)P(B)  [since A and B are independent events]
  • P(AnB) = 0  [since A and B are mutually exclusive events]

Combining the two equations, we get:

  • P(A)P(B) = 0

which means that either P(A) = 0, P(B) = 0, or both have a probability of zero.

Let’s look at an example of events that are independent but not mutually exclusive.

Example: Independent Events That Are Not Mutually Exclusive

In some situations, independent events can occur at the same time.

For example, if we have these events:

  • A is the event that we flip heads on a quarter
  • B is the events that we flip tails on a nickel

The two events are independent, but both can occur at the same time, so they are not mutually exclusive.

Does Mutually Exclusive Imply Independent Events?

Mutually exclusive does not imply independent events.  In fact, if two events A and B are mutually exclusive, then they are dependent.

As explained earlier, the outcome of A affects the outcome of B: if A happens, B cannot happen (and if B happens, A cannot happen).

Is Rolling Dice Independent Events?

Rolling dice are independent events, since the outcome of one die roll does not affect the outcome of a 2nd, 3rd, or any future die roll.

Below, you can see the table of outcomes for rolling two 6-sided dice.  The probability of each outcome is 1/36, which comes from (1/6)*(1/6), or the product of the outcome for each individual die roll.

probability table sum of two 6 sided dice (SHOWING ORDERED PAIRS)
This probability chart shows the ordered pairs (RED, BLUE) when you roll two 6-sided dice (RED is in the first column in bold, and BLUE is in the first row in bold).

(It may help to think of the dice as having different colors – for example, red and blue).

Do Independent Events Add Up To 1?

Independent events do not always add up to 1, but it may happen in some cases. Remember that the probability of an event can never be greater than 1.

Example 1: Independent Events That Add Up To 1

Let’s say you have a quarter and a nickel.  Both are coins with two sides: heads and tails.

We are going to flip both coins, but first, let’s define the following events:

  • A is the event that we flip heads on the quarter.
  • B is the event that we flip tails on the nickel.

These two events are independent, since the outcome of one coin flip does not affect the outcome of the other.

The probabilities are as follows:

  • P(A) = 1/2
  • P(B) = 1/2

So, the probabilities of two independent events add up to 1 in this case: (1/2) + (1/2) = 1.

Example 2: Independent Events That Do Not Add Up To 1

Let’s say that you are flipping a fair coin and rolling a fair 6-sided die.  Let’s define these events:

  • A is the event that you flip heads on the coin.
  • B is the event that you roll 6 on the die.

These events are independent, since the coin flip does not affect the die roll, and the die roll does not affect the coin flip.

The probabilities are as follows:

  • P(A) = 1/2
  • P(B) = 1/6

So, the probabilities of two independent events do add up to 1 in this case: (1/2) + (1/6) = 2/3.

Conclusion

Now you know about the differences between independent and mutually exclusive events.  You also know the answers to some common questions about these terms.

You can learn about real life uses of probability in my article here.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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