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.
Probability: The likeliness that an event will occur. It is measured by the favorable outcomes over the total 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.
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.
What is the probability of flipping a tail and rolling an even number?
Let’s start with the probability of flipping a tail.
Next, what is the probability of rolling an even number?
To find the probability of these independent events, the probabilities are multiplied.
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.
What is the probability of choosing an ace or a queen from a standard deck of cards?
You can also look at the number of favorable outcomes, 8, over the total number of cards, 52. This will also reduce to become
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?
Start with H and compare it to the other 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
Are the following mutually exclusive events?
- Pulling one card and getting a King and Ace from a standard deck of cards
- Rolling a dice and getting an even and the number 6
- Pulling an Ace and a diamond from a standard deck of cards.
3. You can choose an Ace of diamond
Are the following independent events?
1. Flipping a coin and spinning a spinner
2. Choosing a marble from a bag, not replacing it, and then choosing another marble.
2. Since the marble is not replaced, the total number of outcomes has
changed, this affects the second marble.
Three cards, the queen of hearts, 5 of diamonds, and queen of spades, were chosen
from a standard deck of cards. Can you tell if these events were dependent or
We do not know if the cards were replaced after being chosen.
Three cards, the queen of spades, 5 of diamonds, and queen of spades, were chosen
from a standard deck of cards. Can you tell if these events were dependent or independent?
These events were independent.
The repeated choosing of the queen of spades shows that the cards were replaced after being chosen. This means that the outcome of one event did not affect the outcome of the other events.
Given a standard deck of cards.
- Let H = choosing a spade
- Let E = choosing an odd number
- Let F = choosing a eight
- Let A = choosing a King
Which combination of two events are mutually exclusive events?
E and F
E and A
F and A
A card can be both spade and odd, or spade and an eight, or spade and a king.
Any combination with H is not mutually exclusive.
Which of the following is mutually exclusive?
Choice a is mutually exclusive.
Notice that the probability is looking for a or b. This means that there are two different outcomes. Choice b is looking for and which means that the events happen at the same time.
Frequently Asked Questions
The capital P means probability. What is inside of the parenthesis is the event that is the favorable outcome for the experiment. This notation is asking, “what is the probability of getting tails.”
Mutually exclusive means that two events cannot happen at the same time. You can’t make both a right turn and a left turn at the same time. You can’t roll both an even and an odd number on a single die at the same time.
Independent events are two events in which the outcome of one event doesn’t affect the outcome of another event. If you flip two coins, whatever the first coin landed on will not affect how the second coin will land.
The probability of an event is found by creating a ratio of the number favorable outcomes over the total number of outcomes.
In order to tell if an event is dependent is to see if the outcome of the first event affects the outcome of the second event. Are items chosen and then not replaced? This is an indication of dependent events because the total number of outcomes changes from the first event to the second event. If a card is removed from a deck and not replaced the total number of cards went from 52 to 51. That changes the second outcomes probability.
The sample space is the set of all of the possible outcomes. This is often a list, not a number. The sample space for flipping a coin is heads or tails,