What are the chances of getting an A in math class and no homework assignments today? Or that you’ll score the winning goal in the soccer match and your crush gives you a thumbs-up?
Sounds like a super lucky day, right?
That kind of double-luck is something you can actually calculate with math.
In this fun, kid-friendly guide, we’ll explore the probability of two events happening using simple explanations, relatable examples, a fun quiz, and answers to common questions.
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Probability is the math we use to figure out how likely something is to happen. It helps us make predictions and smart decisions every day, even when we don’t realize it.
We write it as a number between 0 and 1.
0 means the event is impossible.
1 means it’s certain to happen.
Anything in between? That’s a chance somewhere between “no way” and “for sure.”
We use this formula to calculate probability:
P(Event) = \(\displaystyle \frac{\mathrm{Number\ of\ favorable\ outcomes}}{\mathrm{Total\ number\ of\ possible\ outcomes}}\)
Let’s look at a quick example.
What’s the chance of getting heads when you flip a coin?
You’ve got two sides—heads and tails—so the chance of getting heads is \(\displaystyle \frac{1}{2}\). That’s right in the middle of 0 and 1.
P(Heads) = \(\displaystyle \frac{1}{2}\)
In probability, an event is anything that might happen.
Rolling a die and getting a 3? That’s an event.
Picking a red marble from a bag? Another event.
Even something like getting extra recess is an event if there’s a chance of it happening.
Let’s try a quick check:
What’s the probability of rolling a 3 on a regular six-sided die?
P(3) = \(\displaystyle \frac{1}{6}\)
There’s one 3 out of six possible numbers, so the answer is 1 out of 6, or \(/displaystyle\frac{1}{6}\)
You May Also Like: The Multiplication Rule of Probability
So far, we’ve looked at finding the chance of one event happening.
But what if you want to know the chances of two things happening together?
Let’s say you’re crossing your fingers for a field trip this weekend and sunny weather. What are the chances that both of those actually happen?
This is where we use something called the probability of two events happening. It tells us how likely it is that two separate events both occur at some time.
Here’s the thing: how you calculate the probability of two events happening depends on the relationship between the two events.
There are three different ways events can be connected:
Two events are independent if the outcome of one doesn’t affect the outcome of the other.
For example, flipping heads of a coin and drawing a queen card from a deck are independent; the coin flip does not change the card draw.
To determine the probability of two independent events, P(A) and P(B), both occurring, we multiply the probabilities of each event together:
P(A and B) = P(A) × P(B)
For example: What’s the chance of flipping heads (\(\displaystyle \frac{1}{2}\)) and picking a queen from a standard deck of 52 cards (\(\displaystyle\frac{1}{52}\))?
Since flipping a coin does not impact the card you’ll pick from a deck (and vice verse), these two events are independent, so we’ll calculate probability with a simple multiplication:
P(Heads and Queen) = \(\displaystyle \frac{1}{2} \times\frac{1}{52} = \frac{1}{104}\)
Dependent events are events where the outcome of the first changes the probability of the second. That change usually happens because something was removed, used up, or affected by the first outcome.
Imagine a bag with 5 pieces of candy: 2 cherry and 3 lemon flavored ones.
Let’s find the probability of picking two cherry candies in a row, without putting the first one back in the bag. This is a dependent event because our chances of picking the cherry flavor change after we take one out. The first time we go for a cherry flavor, the ratio is 2 cherries to 3 lemons. If we take one cherry out, the ratio of candies in the bag becomes 1 cherry to 3 lemons.
We’ll use the formula for dependent events to find out what the chances are of picking two cherries in a row:
P(A and B) = P(A) × P(B after A happens)
We read this formula as:
The chance of both things happening equals the chance of the first thing × the chance of the second thing after the first one already happened.
Now, let’s find P(A), where (A) represents our chances of picking a cherry first from the bag.
There are 2 cherry-favored candies and 5 candies in total in the bag.
So, your chance of picking a cherry first is:
P(1st Cherry) = \(\displaystyle\frac{2}{5}\)
Now, imagine you have already picked a cherry. That means:
Only 1 cherry remains.
There are now 4 candies left in the bag.
Let’s find P(B), which represents our 2nd cherry.
P(1st Cherry) = \(\displaystyle\frac{1}{4}\)
The chance of picking two cherries in a row would be:
P(1st Cherry and 2nd Cherry) = \(\displaystyle\frac{2}{5} \times \frac{1}{4} = \frac{1}{10}\)
So there’s a 1 in 10 chance of picking two cherry candies in a row, and we found that using the rule for multiplying dependent probabilities.
Mutually exclusive events are events that cannot happen at the same time. If one happens, the other must be ruled out.
Imagine you roll a six-sided die. You might get a 2, or you might get a 5—but you can't get both on a single roll. Or, think about drawing one card from a deck of playing cards. If that card is a heart, it can’t also be a club.
You can also picture this with everyday activities.
Say you’re deciding whether to go to a basketball game or a movie on Friday night, and both start at the same time. If you attend the basketball game, you can’t be at the movie theater at the same time. You have to choose one or the other—those events are mutually exclusive.
Each of these cases involves events that cancel each other out when one occurs.
When two events are mutually exclusive, we calculate the probability that one or the other will happen by adding their individual probabilities together.
The formula looks like this:
P(A or B) = P(A) + P(B)
This only works when the two events cannot happen at the same time.
Let’s go back to the die example. You roll a standard six-sided die. What’s the probability that it lands on either a 2 or a 5?
First, calculate the probability of each outcome. The chance of rolling a 2 is one out of six, or \(\displaystyle\frac{1}{6}\). The chance of rolling a 5 is also \(\displaystyle\frac{1}{6}\).
Since these are mutually exclusive events, and only one can happen in a single roll, we add the two probabilities using the formula:
P(A or B)=P(A) + P(B)
P(2 or 5) = \(\displaystyle\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\)
So, you have a one-in-three chance of rolling a 2 or a 5 on a single roll.
You’re not trying to find the chance of both events happening together—you’re asking, “What are the chances I get one or the other?” Since there’s no overlap (the die can’t show two numbers at once), you simply add the chances together.
That’s the key to understanding mutually exclusive events: they are separate paths, and you’re choosing between them, not combining them.
Now that you know how to work with different types of events—independent, dependent, and mutually exclusive—let’s put your skills to the test.
Try these problems on your own first, then check the answers below to see how you did!
You have a 1 in 2 chance of rain today, and a 3 in 5 chance of wearing your favorite sneakers. What’s the probability that both happen on the same day?
You draw two marbles from a bag that contains 3 blue and 2 red marbles, without replacing the first. What’s the probability that you draw two blue marbles in a row?
What’s the probability of rolling a 1 or a 6 on a standard six-sided die?
You flip a coin to decide who picks the movie, and then you randomly choose a snack from six options. What’s the chance the coin lands on heads and you pick popcorn?
You draw one card from a regular deck. What’s the probability it’s a Queen or a King?
What’s the chance of flipping heads on a coin and rolling a 4 on a six-sided die?
Here are some common questions we hear at Mathnasium of Cherry Hills, along with answers to help you master the probability of two events happening.
When events are independent, both can happen, so we multiply. When they’re mutually exclusive, only one can happen, so we add.
Nope! Probability always stays between 0 (impossible) and 1 (certain). Anything more than 1 means something went wrong in the math.
That’s okay! Small numbers mean the event is unlikely, but still possible. A 1 in 100 chance means it could happen, just not often.
No. Mutually exclusive means one event happening completely prevents the other. They can’t influence each other—they cancel each other out.
It always adds up to 1. That’s because something has to happen.
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