Probability is something that we deal with every single day without realizing it. Whenever we open our weather app to check for the chance of rain, or when we consider the possibility of winning the lottery, we're really dealing with probability. But probability isn't just something that we can think about or consider; it's something that we can actually calculate. Now I know this might not be the best news for you, knowing that you have another thing to calculate, but you don't have to worry because here I'm going to walk you through the basics of calculating probability and some of the notation that goes along with that. So let's go ahead and get started.
Now, when working with probability mathematically, we consider the probability of different events, and we write this as \( p \) with the event in parentheses. Now, when I say event, I don't just mean some big event like a party; an event is anything, no matter how big or small, that could happen. So, if I was considering the probability of it raining, I would write \( p(\text{rain}) \). Or if I was considering the probability of getting heads when flipping a coin, I would write \( p(\text{heads}) \). Now, when actually calculating probability, we need to consider the two different types.
So, if I have a coin in my hand and I'm going to flip it but I want to know the probability of getting heads if I do, since this is based on what could possibly happen, this is referred to as theoretical probability. But if I were instead to take that coin and flip it three times and record the results of each flip and calculate the probability of getting heads based on that, since this is based on what actually did happen, this is referred to as empirical or experimental probability since we performed an experiment here. Now, when calculating the theoretical probability of getting heads, since I know there are only two possibilities when I flip a coin, heads or tails, and only one of those is heads, the theoretical probability of getting heads on a coin flip is equal to one-half. Now with our empirical probability over here, since we got heads on two out of those three tosses, the empirical probability of getting heads on a coin flip is equal to two-thirds. Now the most important thing to consider here is that our theoretical probability was calculated before any events actually happened, whereas our empirical probability was calculated after our events actually occurred based on data.
Now when looking at our formulas that we actually used to calculate these here, these look slightly different just because they're tailored to each different type of probability, but they're really the same thing. So with our theoretical probability here, we took the number of outcomes that included our event. So in this case, there was only one way to get heads, and then we divided that by the number of total possible outcomes, either heads or tails. But with our empirical probability, we took the number of times our event actually occurred, in this case, twice, and divided that by the number of total trials.
So, we still took our event and divided it by our total. Now that we've seen the basics of calculating probability, let's work through some examples together. So, looking at our first example here, we're asked when rolling a six-sided die, what is the probability of rolling a number greater than three? So here we're looking for the probability of getting a number greater than three. Now when rolling a six-sided die, I know that I could roll any number one through six.
And when looking at these possible outcomes here, I see that only three of these numbers are greater than three. So when calculating the probability here, I would take the possible outcomes here that include my event, in this case, three of them, and divide it by the number of total possible outcomes. In this case, since there are six sides of that die, there are six total possible outcomes. Now simplifying this fraction gives me a probability of one-half, and you'll often see probabilities reported as decimals. So here the probability of rolling a number greater than three is equal to 0.5.
Let's look at our other example here. Here we're given data and asked to find the probability of rolling a number greater than three based on that data of rolling a six-sided die ten times. So here we're calculating empirical probability because we're actually given data to work with of events that already happened. So here we're calculating still the probability of rolling a number greater than three, but we're going to take a look at this data table. So since there were ten total rolls here, I already know that the number on the bottom of that fraction is going to be ten for the total number of trials.
Then I want to look for the number of times that my event actually occurred. Since my event was rolling a number greater than three, I want to look for the number of times that that happened. So on that first roll, I got a six, then I got a four, then I got a bunch of fives in a row and a six and then a four and a five. So eight out of those ten times, I rolled the number greater than three. So taking that probability and simplifying that fraction, I get a probability of four-fifths or as a decimal, 0.8.
Now something that you might be wondering here is why our empirical probability is so different than our theoretical probability when we're technically calculating the probability of the same event. And the answer comes down to our sample size. So since we only did ten trials, this is not quite so close to our theoretical probability. But if I were to roll the dice 100 or 1,000 times, I would get closer and closer to that theoretical probability of 0.5. So this is something to consider in your science courses when you're working with your own data.
Now when you're working through probability problems, you may see all of the possible events, possible outcomes of an event expressed as a set, so in curly brackets from set notation. Now this will be referred to as a sample space, and it just shows all of the possible outcomes in between those curly brackets. So for example, the sample space of flipping a coin, I would say that \( S \) equals, \( S \) is the sample space, and then I would put all the possible outcomes in between those curly brackets, in this case, either heads or tails. So this here represents the sample space of flipping a coin. Now that we've seen the basics of calculating probability and we're more familiar with some of that notation, let's get some more practice.
Thanks for watching, and I'll see you in the next one.
