Hey, everyone. So far, we've learned how to calculate the probability of one event or another event happening, like, say, getting heads or tails on a single coin flip. But what if someone asked you to calculate the probability of getting heads on one coin flip and then getting tails on another coin flip? Well, it turns out that this is actually an entirely different type of probability that you might hear referred to as and probability. Now we're going to have to calculate this and probability, but before we dive too deep into that, we're going to need to talk about what it means for events to be independent from each other.
Now I know that that might sound like a lot, but here I'm going to walk you through exactly what it means for 2 events to be independent or dependent and then how to calculate the and probability of 2 events that are independent. So let's go ahead and get started. So when considering 2 different events, if these two events do not depend on each other, they are referred to as being independent. So in other words, the outcome of one event has no effect whatsoever on the outcome of the other event. Now this can seem a little bit abstract, so let's take a look at some different sets of events and determine whether or not they're independent.
So looking at my first example here, I have getting tails on the first toss of a coin and then getting tails on the second toss of a coin. Well, what happens in the first toss has no effect whatsoever on what happens in the second toss, so that tells me that these events are independent from each other. What happens in one doesn't affect what happens in the other. Let's look at another set of events here. We have drawing and keeping a blue marble from a bag and then drawing a blue marble again.
Well, if I consider what's actually happening here, if I reach into that bag and I draw a blue marble on that first draw and I keep it in my hand and then I go back in the bag and I want to know the probability of drawing a blue marble again, well, I've already removed one of the blue marbles so my chances have lessened because of what happened in that first event. Now because what happens in the first event affects what happens in the second, these two events are not independent. They are dependent because what happens in one depends on what happens in the other. So now that we've seen how to determine whether events are independent or dependent, let's take a closer look at our independent events here. So for 2 independent events, we're going to need to calculate the probability of event a and event b occurring.
Now earlier, I said that you might hear this referred to as and probability, and in set notation, you'll see this denoted with this little upside down u symbol that just means and. Now actually, calculating this probability is super simple because all we're going to do is multiply our probabilities together. So we're going to take the probability of event a occurring and simply multiply it by the probability of event b occurring in order to get the probability of a and b occurring. So now that we've seen this formula, let's go ahead and apply it to some examples that we'll work through together. So looking at our first example here, we want to find the probability of getting heads on 2 consecutive coin flips, so the probability of getting heads and then getting heads again on a second flip.
So in order to do this, we're just going to take these individual probabilities and multiply them together. So we're going to take the probability of getting heads on that first flip and then multiply it by the probability of getting heads on the second flip. So looking at these probabilities, I know that there are only 2 possibilities when flipping a coin, either heads or tails, and only one of those is getting heads. So here the probability of getting heads on that first coin flip is 1 over 2. Now I'm going to multiply this by the probability of getting heads on the second coin flip which is actually going to be the exact same thing because again there are still 2 possible outcomes getting heads or tails and only one of those is getting heads.
So I'm going to take that 1 half and multiply it 1 by 1 half again. Now when we multiply across here I'm going to end up getting 1 fourth or as a decimal 0.25 as my probability of getting heads and then getting heads again. Now we can actually reason this out because there's not a ton of possibilities here, right? So if I consider the 1st flip of a coin and the 2nd flip of a coin, on that first flip, I could get heads. Then on my second flip, I could get heads again, or I could get heads on that first coin flip and then get tails on the second.
But I could also get tails on the first flip and then get heads on the second, or I could get tails on the first flip and tails on the second. Now these are the only 4 possible things that could happen and only one out of these fours is getting heads on 2 consecutive coin flips. Now this backs up what we just calculated because we found that there is a 1 in 4 chance of getting heads on 2 consecutive coin flips. Now let's take a look at another example here. We have rolling an even number on the first roll of a 6 sided die and then rolling a 3 on the 2nd roll.
So we're trying to find the probability of getting an even number and then getting a 3 when rolling this die. So of course, we're going to take these individual probabilities and multiply them together. So we're going to take the probability of getting an even number and then multiply it by the probability of getting a 3. Now when rolling a 6 sided die, I know that there are 6 total possibilities. And in order to roll an even number, I could either get a 2, a 4, or a 6, and those would all be even numbers.
So that tells me since there are 3 possible ways to get an even number that my probability of getting an even number at all is 3 out of those 6 total possibilities. Now I'm gonna multiply this by the probability of rolling a 3, which there is only one way to roll a 3, so I know that this probability is 1 over 6. Again, multiplying across here, this is going to give me 3 over 36, which as a decimal is simply 0.08, so not a very high chance that I'll roll an even number and then roll a 3. Now there's one final thing that I want to mention if you ever run into a problem that asks you to find the and probability of more than 2 independent events. And it's super simple.
All you're going to do is keep multiplying all of those probabilities together to find the probability of a and b and c occurring. Just keep multiplying those individual probabilities. Now that we know how to find the and probability of multiple independent events, let's get some more practice. Thanks for watching and I'll see you in the next one.
