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Probability of Mutually Exclusive Events

Hey, everyone. So far, we've been dealing with the probability of individual events, like say the probability of picking a blue shirt out of my closet to wear on a given day. But what about the probability of multiple events? And further than that, what about multiple events that can happen at the same time, like, say, wearing a blue shirt with a green pair of pants, versus those that cannot, like, say, wearing a blue shirt and wearing a green one too? Well, this might sound like it's going to be complicated, but here I'm going to break it down for you and show you exactly what the difference is between events that can happen at the same time versus those that can't so that we can calculate the probability of those that can't.

So let's go ahead and get started here. Now events that cannot happen at the same time actually have their own special name, and they're referred to as being mutually exclusive. Now you might see multiple events represented using a circle diagram like this one that we see here. Here, we have event a, wearing a blue shirt and event b, wearing a green shirt. Now in this circle diagram, you might notice that these are two completely separate circles.

I have event a and event b, but there's no region of overlap. Because these events are completely separate, they cannot happen at the same time. It's either wear a blue shirt or wear a green shirt, but never both. Now because of that, these events can't happen at the same time. They're referred to as being mutually exclusive.

They're completely exclusive from one another. Now over here, we have event a as still wearing a blue shirt, but we have event b as wearing green pants. Now in this circle diagram, we see this region of overlap in the middle and this represents both of these events happening together, wearing a blue shirt with a green pair of pants. So because of this overlap, we see that these events can happen at the same time and that means that they are not mutually exclusive. Now we're going to have to differentiate between sets of events that are mutually exclusive versus those that are not.

So let's get a bit more practice with that now that we know what mutual exclusivity even is. So looking at this set of examples here, I see getting heads when flipping a coin versus getting tails. Now whenever you flip a coin, we know that there are only two things that could happen. You could either get heads or tails but can never get both at the same time on a single coin flip. Now because these can't happen at the same time, these are mutually exclusive.

They're completely exclusive from one another, and their circle diagram would look similar to this one up here. Now let's look at our second set of events here. We have getting a 6 when rolling a die versus getting a number higher than 3. Now let's think about this a little bit. When we roll a six, there's only one way to do that, by rolling a six.

But if we want to roll a number higher than three, we could get a 4, a 5, or a 6. There are three possible outcomes here. Now one of those outcomes is getting a six, which is exactly what my other event is. So if you were to roll a six, you would be satisfying both of those events and they would be happening at the same time.

Now because of that, these events are not mutually exclusive. Now here we want to dive deeper into our events that are mutually exclusive and calculate the probability of some event a, or some event b occurring. So let's go ahead and dive deeper into that here. So to find the probability of any one of multiple mutually exclusive events occurring, all we're going to do is add the probabilities of each of them. So I'm going to take the probability of event a and add it together with the probability of some event b.

And that's all. This will give me the probability of my two events, a or b, occurring. Now this is sometimes referred to as or probability because it's a or b, but never both, right, because they can't happen at the same time. And you might see this with this little u symbol that just means or in set notation. So now that we've seen this formula, let's go ahead and apply it to an example.

Here, we roll a six-sided die, and we want to know the probability of getting a 3 or getting a 5. So I have two events here. I'm rolling a 3, or rolling a 5. So here, I want to calculate the probability of rolling a 3 or rolling a 5. So with this color-coded here, we see our two events, 3 or 5.

And in order to calculate this probability, we saw that we just need to go ahead and add these together. So I'm going to take the probability of rolling a 3, and then I'm simply going to add it together with the probability of rolling a 5. So we just need to look at these individual probabilities to come to our final answer here. So I know that whenever I roll a six-sided die, there's only one way that I could roll a 3 by just rolling that.

Right? So there's one way to get to that outcome, and there are six total possible outcomes here because it's a six-sided die. Now I'm going to take that probability and add it together with my probability of rolling a 5. Now I know that there's also only one way to roll a 5 here, so this probability will actually end up being the same. So I have one-sixth plus one-sixth.

Now actually adding these two together gives me a value of 2 over 6 or, as a simplified fraction, one-third. Now we know that one third as a decimal is equal to around 0.33, and that would go on and on. So our probability here of rolling a 3 or rolling a 5 is equal to one-third, or 0.33. Now that we know what or probability is and how to calculate it for mutually exclusive events, let's get some more practice. Thanks for watching, and I'll see you in the next one.

If a single card is randomly selected from a deck of cards, what is the probability of selecting an ace or a king?

• A. $0.0059$
• B. $0.077$
• C. $0.15$
• D. $0.85$

For two mutually exclusive events A and B, compute $P\left(A\cup B\right)$ if $P\left(A\right)=0.15$ and $P\left(B\right)=0.32$

• A. $0.048$
• B. $0.17$
• C. $0.47$
• D. $0.53$

Probability of Non-Mutually Exclusive Events

Hey everyone. We just learned how to calculate the "or" probability of 2 mutually exclusive events by simply adding the individual probabilities together. But what about the "or" probability for 2 non-mutually exclusive events? Well, you may be worried that we're going to have to learn a brand new formula here, but you don't have to worry about that because here, I'm going to walk you through how calculating the "or" probability for non-mutually exclusive events is actually almost identical to calculating it for mutually exclusive events, just with one extra step added in. So let's go ahead and get started so that then you can calculate the "or" probability for any events.

So looking at this, remember for our non-mutually exclusive events, there exists this region of overlap in the middle where both of these events are happening at the same time. Event a and event b are happening here. Now in set notation, you'll see this written as a and b with this little upside-down U symbol that just means "and". Now in actually calculating the probability of a or b happening for these non-mutually exclusive events, we're going to start out the same exact way we did for our mutually exclusive events. We're going to take the probability of our event a, in this case, wearing a blue shirt, whether it be with green pants or not, and we're going to add in our probability of event b, in this case, wearing green pants.

But in adding in our probability of wearing green pants, we already accounted for the time that it got worn with our blue shirt. So wouldn't we be counting it twice here? Well, we would be counting it twice. So we really just want to add in the time where we're wearing green pants without a blue shirt. But how do we do this?

Well, in order to get rid of that extra outcome and not count it twice, we need to subtract the probability of that overlap region, the probability of a and the probability of b, to only have counted that area once. So in order to calculate the probability of a or b occurring, we're still adding the probability of a plus the probability of b, but now we're just subtracting the probability of a and b. Now, this might seem a little abstract and overwhelming seeing this equation for the first time, but here we're going to walk through an example together. So let's go ahead and take a look at this example down here, and we'll see exactly how this equation works. So here we see when rolling a 6-sided die, what is the probability of rolling a number greater than 3 or an even number?

So here we have 2 events, events, rolling a number greater than 3, or rolling an even number. So when rolling a 6-sided die, let's think through all of our possible outcomes here. Well, we could roll a 1, but that actually isn't a part of either of our events. We could also roll a 2. Now a 2 is even, but it's not greater than 3.

We could also roll a 3, which is not a part of any of my events either. Then I could roll a 4. Now a 4 is both even and greater than 3, so it's going to go in this middle region here. Then I could roll a 5. A 5 is not an even number, but it is greater than 3.

Then finally, I could roll a 6, which again is both greater than 3 and even, so it's going to go in this overlap region here. Now from here, let's go ahead and get into calculating this probability, the probability that we'll roll a number greater than 3 or an even number. Now in doing this, remember, we're going to take those individual probabilities and add them together to start out here. So we're going to take the probability of rolling a number greater than 3. Looking at my circle diagram here, I see that there are 3 possible outcomes, either a 4, a 5, or a 6, out of 6 total possible outcomes.

So that's my first probability. Then for my second event, rolling an even number, I could roll a 2, a 4, or a 6 out of 6 total possible outcomes again. So once again, we get 3 out of 6 here. But now I need to subtract that region of overlap where my numbers are both greater than 3 and even. And looking at my diagram here, there are 2 outcomes in which that would happen, rolling a 4 or rolling a 6.

So here, I'm going to take that 2 out of 6 total possible outcomes. Now from here, all that's left to do is addition and subtraction. So first, taking that addition, 36 plus 36 is going to give me 66. And then I'm subtracting that 26. Now subtracting that gives me 46, or as a simplified fraction, 23.

Now, as a decimal, if you want to express your probability here as a decimal, this gives us 0.67. So the probability of rolling a number greater than 3 or an even number is equal to 0.67. Now, this makes sense, right, because looking at this number, 4 out of 6, and comparing that to our circle diagram, there are 4 possibilities that are part of both of these events. So a 5, a 4, a 6, or a 2, those are 4 things that are either greater than 3 or an even number, maybe both mixed in there, and it's 6 total outcomes. So 4 out of 6 makes perfect sense here.

Now having calculated the "or" probability for mutually exclusive and non-mutually exclusive events, it may seem like there are 2 separate formulas to remember here, but not really because this orange formula here is actually the most general way to calculate the probability of any 2 events, a or b, whether they're mutually exclusive or not. But if they are mutually exclusive, this last term is just going to be 0. So this formula will end up looking like this for our mutually exclusive events. So the equation for the probability of a or b is really the same regardless of mutual exclusivity. But for our mutually exclusive events, the probability of a and b is simply always going to be 0.

So now that we know how to find the "or" probability for any 2 events, mutually exclusive or not, let's get some more practice. Thanks for watching, and let me know if you have questions.

Probability of Non-Mutually Exclusive Events Example

Hey, everyone. In this problem, we're given a data table and we're told that this table shows the outfits of 300 observed people on a given day. And we want to know, of 1 person randomly selected from this group of 300 people, what is the probability that they will either be wearing shorts or wearing a green shirt? Now this might look overwhelming just because there's a lot of data there, but let's go ahead and interpret it together. So we see that of the people that are wearing shorts, we come to a total of 188 out of those 300 people that are wearing shorts.

Now our second event is wearing a green shirt. So we have 106 total people that are wearing a green shirt, and we can see that breakdown. Now this is actually a great way to visualize non-mutually exclusive events because we see this region of overlap here where both of these things are happening. People are wearing a green shirt with shorts, so both of these events happen together. Now whenever we're dealing with events that are non-mutually exclusive, we know that we need to account for that region of overlap when calculating the probability.

So we know that the probability of our events, a or b happening, is equal to adding those 2 events together, so taking the probability of a, in this case, we'll refer to wearing shorts as event a, and then adding that with the probability of event b, in this case wearing a green shirt, and then subtracting that region of overlap. So subtracting the probability of a and b happening, just like we see in our table here. We see that people are doing both event a and event b. Now we can go ahead and use our data to fill in the gaps here and calculate our final probability. So for the probability of event a, we know that there are 188 out of 300 total people wearing shorts.

So we want to take that 188, put it over a total of 300. Then we want to add that together with our probability of event b. So we see that there are 106 total people wearing a green shirt. So we can take that 106 and again put it over our total of 300. Then finally, we want to subtract that region of overlap, the people that are doing both of these events, wearing shorts with a green shirt.

So we're going to take that number of 89 and put it over our total as well, so 89300. Now from here, we're just left to do this calculation. So I can go ahead and add these two values together, and that gives me a number of 294300. That's a lot of people that are doing both of those events, but I need to subtract that region of overlap. So subtracting that 89300 and doing this subtraction, I end up with a value of 205300.

Still a significant amount of people, but not quite so many. Now we can go ahead and reduce this fraction to 4160 or if you want to express this as a decimal, this goes down to 0.68 as our final answer. So of 1 randomly selected person of this group of 300, the probability that they will be wearing shorts or a green shirt, maybe even both, is 0.68. Thanks for watching and I'll see you in the next one.

A card is drawn from a standard deck of 52 cards. What is the probability that the card is a diamond or a king?

• A. $0.33$
• B. $0.31$
• C. $0.15$
• D. $0.85$