Independent VS. Mutually Exclusive Events

Probability and chance surround us every day. If you have ever watched the news, the weather and market trends are given as probabilities. The chance of rain or snow is a great example of probability in everyday life.

Before going over independent and mutually exclusive events, let’s review some important probability vocabulary.

Vocabulary

Probability: The likeliness that an event will occur. It is measured by the favorable outcomes over the total possible outcomes.

$$P\;\left(event\right)\;=\;\frac{Number\;of\;favorable\;outcome}{Total\;number\;of\;possible\;outcomes}$$

Event: An event is an outcome or a set of outcomes of an experiment.
Sample Space: The set of all possible outcomes of an experiment.

Types of Events

A mutually exclusive event is a situation in which two events cannot occur at the same time.

Independent events are when one event is unaffected by another event.
Dependent events are when one event affects the outcome of another event.

Simple: An event with a single outcome
Compound: Two independent events occurring. Found by multiplying the probabilities of the events.

Let’s take a look at flipping a coin. When you flip a coin there are two possible outcomes. One outcome is the coin lands on heads, the second outcome is that the coin lands on tails.

The experiment is flipping the coin. The sample space for this experiment is heads and tails.

When you flip a coin once you cannot get both heads and tails. This means that the events of getting heads or getting tails are mutually exclusive.

This is also a simple event since there is one event that is taking place.

Notice that with the diagram, there is no overlap between the two events.

Now let’s say you are going to roll a standard six sided die and flip a coin.

Since there are two events this is now a compound event. If you land on four with the die, that does not affect what the coin will land on. These two events are independent.

Keeping with the numbered die, two events that would not be mutually exclusive would be rolling an odd number and rolling a 5.

Rolling a 3 would satisfy one of the events, rolling an odd. However, if a 5 is rolled, it satisfies both events and happens at the same time.

Notice the overlap with these events. There are numbers that are odd that are not five, but a five is always odd.

Did you know?

• Mutually exclusive events are always dependent.
• The probability of two people in a room of 23 people having the same birthday is 50%
• A probability of 0 means that the event is impossible
• A probability of 1 means that the event is a certainty.

Probability Notation

What is the probability of rolling a 4 with a standard six sided die? Let’s start by looking at the sample space, the different numbers that the die can land on.

Sample Space: 1, 2, 3, 4, 5, 6

The favorable outcome for this experiment is 4.

$$\operatorname{𝑃}\;\left(event\right)\;=\;\frac{Number\;of\;favorable\;outcome}{Total\;number\;of\;possible\;outcomes}$$

$$\operatorname{𝑃}\;(rolling\;\operatorname{𝑎}\;4)\;=\;\frac16$$

Example 1:

What is the probability of flipping a tail and rolling an even number?

$$\operatorname{𝑃}\;(tails\;and\;even)\;=\;?$$

$$P\;(tails)\;=\;\frac1{\;2}$$

Next, what is the probability of rolling an even number?

$$P\;\left(even\right)\;=\;\frac3{\;6}\;=\;\frac{\;1\;}2$$

To find the probability of these independent events, the probabilities are multiplied.

$$P\;\left(tails\;and\;even\right)\;=\;\frac1{\;2}\;\times\;\frac1{\;2}\;=\;\frac1{\;4}$$

Example 2:

What is the probability of flipping both a tail and heads with one coin?

Since only one coin is being used, you can only get one result, this means that the events are mutually exclusive.

$$P\;\left(heads\;and\;tails\right)\;=\;0$$

Example 3:

What is the probability of choosing an ace or a queen from a standard deck of cards?

$$P\;\left(Ace\right)\;=\;\frac4{52\;}\;=\;\frac1{13}$$

$$P\;\left(Queen\right)\;=\;\frac4{52}\;=\;\frac1{13}$$

$$P\;\left(Ace\;OR\;Queen\right)\;=\;\frac1{13}\;+\;\frac1{13}\;=\;\frac2{13}$$

You can also look at the number of favorable outcomes, 8, over the total number of cards, 52. This will also reduce to become $$\frac2{13}$$ .

Example 4:

Given a standard deck of cards.
Let H = choosing a heart
Let E = choosing an even number
Let F = choosing a five
Let A = choosing an Ace

Which of the following are mutually exclusive events?

• Can a card be both a heart and an even? Yes
• Can a card be both a heart and a five? Yes
• Can a card be both a heart and an Ace? Yes

Next look at E.

• Can a card be both an even and a five? No- mutually exclusive events
• Can a card be both an even and an Ace? No – mutually exclusive events

The last pairing is F and A

• Can a card be both a five and an ace? No – mutually exclusive events