An Intro to Probability | Basic Probability | Math with Mr. J

[Music] Welcome to Math with Mr. J. In this video, I'm going to go through an introduction to probability. We will start with two basic probability intro examples where we will really break down the probability formula and finding the probability of something. We will then move on to our second section where we will go through some more probability examples using playing cards. And then lastly our third section we will talk about the probability line which will help our overall understanding of probability. Now simply put probability tells us how likely something is to happen. In other words the chance of something happening. Not all things in life are sure things. We can't predict everything with 100% certainty. Meaning we may not know what's going to happen. So when we are unsure about the outcome, the result of something, we can use probability to help us understand how likely something is to happen. For example, if we flip a coin, it can be heads or tails. We can't predict with 100% certainty. We don't know for sure, but we can use probability to tell us how likely a certain outcome, a certain result will be. Let's jump into number one where we have a spinner and we need to figure out what's the probability that we spin this spinner once and we land on three. Well, let's break this down and we'll start with the probability formula which is right here. We have P for probability and then in parenthesis we have the word event and an event is whatever we are finding the probability of. Basically, an event can be one or more outcomes that we are interested in and focused on. The probability of an event equals the number of favorable outcomes over the total number of possible outcomes. Favorable outcomes are all of the ways our event can happen. Then the total number of possible outcomes, that's the total number of things that can happen. the total number of possibilities. All of this will start coming together as we go through our examples. Let's start with the number of favorable outcomes we have. How many ways can our event of spinning a three happen? So, we need the number of sections on the spinner with a three. Well, there's one. We have one favorable outcome on that spinner. One way that our event can happen. So, favorable outcomes, one. Next, we have how many total possible outcomes do we have? So, how many total sections can the spinner possibly land on? Well, there's section 1, 2, 3, and four. There are four equal-sized sections that the spinner can land on. So, each section has an equal chance of being landed on. So again there are four total possibilities. Now let's use this information to write out the probability which we are going to write as a fraction decimal and percent. So we have P for probability. And now we need our event in parentheses. That's landing on a three. So let's put a three equals well we need the number of favorable outcomes 1 over the total number of possible outcomes four that's our fraction 1 over 4 1/4 and that's our probability we have a 1 out of four chance of spinning a three. Now let's write that as a decimal and a percent. 1/4 as a decimal is 0.25 25 hundredths. Remember divide the numerator the top number by the denominator the bottom number to go from a fraction to a decimal. And then as far as the percent we get that percent by multiplying the decimal by 100, which a quick way to do that is move the decimal twice to the right. So our percent is 25%. And this is the probability that we spin this spinner once and land on three. This describes our chances. This helps us understand how likely it is that we spin a three. We actually have less than a 50% chance here. One section out of four has a three and that's up against three other sections that have other numbers. The probability of spinning a three here is more unlikely than likely. We have a higher probability, a better chance the spinner lands on one of the other numbers. So, we call this an unlikely event. Now, with that being said, can we still land on a three? Absolutely. I'm not saying we can't. We're just thinking about the likelihood that we do. So, this is something to start thinking about what the probability is telling us. Let's move on to number two. Taking a look at number two, we have what's the probability of picking a red marble. So, let's say we have all of these marbles in a bowl and we randomly select one without looking. So each marble has an equal chance of being selected. What's the probability we pick a red marble? Well, as far as our favorable outcomes, we could pick red marble number one, two, three, four, five, six, or seven. So there are seven red marbles. That means we have seven favorable outcomes. seven ways that we can end up picking a red marble. Now, we need the total number of possible outcomes. So, the total number of marbles. Well, we have one green marble, two green marbles, and then one blue marble. So, seven red marbles plus those other three marbles gives us a total of 10 marbles. So we have the probability and then our event is picking a red marble. So we will put red here equals the number of favorable outcomes that's seven over the number of total possible outcomes that's 10. So we get 7 over 10 7/10ths. That's our fraction. We have a 7 out of 10 chance of picking red. Now let's write this as a decimal and percent. 7/10. As a decimal is 0.7. So 7/10. Now remember, divide the numerator by the denominator to go from a fraction to a decimal. And then as far as our percent, we multiply that decimal by 100. A quick way to do that is to move the decimal twice to the right. Our percent here is 70%. So this is the probability of picking a red marble. This describes our chances. This helps us understand how likely we are to randomly pick a red marble. So thinking about our chances here, we have more than a 50% chance of picking red. We have seven red marbles out of the 10 up against three marbles out of the 10 that are not red. Our event of picking red is more likely to happen than unlikely. We have a higher probability of picking red, a better chance of picking red than picking another color. We call this a likely event. So there's our two introductory examples. Let's move on to our second section. Here are our next probability examples. Let's jump into these examples where we are going to say that the playing cards are face down on a table so we cannot see what they are. We're going to randomly select one. So each card has an equal chance of being selected. We have five events here that we are going to find the probability of. Let's start with number one where we have the probability of picking a nine. So picking a nine is our event. If we randomly pick one card, what's the probability we pick a nine? And we're going to write the probability as a fraction, decimal, and percent. Let's start with the number of favorable outcomes. So how many ways can this event happen? How many cards have a nine? So looking at the cards we can see one. One card has a nine. So our numerator is one. Now we need the total number of possible outcomes. The total number of things that can happen. So how many possibilities do we have here? How many cards are there? Five. So five is our total number of possible outcomes. So five is our denominator. We can pick any one of those five cards. So we have five possible outcomes, five possible results. That's our fraction 1 over five. 1/5. And that's our probability. We have a 1 out of five chance of picking a nine. Now let's write that as a decimal and a percent. 1/5 as a decimal equals 0.2 two 2/10. Remember, divide the numerator, the top number, by the denominator, the bottom number, to go from a fraction to a decimal. And then we can get our percent by multiplying the decimal by 100, which a quick way to do that is move the decimal twice to the right. So our percent here is 20%. So this is the probability of picking a nine. This describes our chances. This helps us understand how likely it is to happen. Let's move on to number two where we have what's the probability of picking an odd number. Let's start with the number of favorable outcomes. So how many cards have an odd number? We have a seven, a five, a nine, a seven, and a three. So all five cards we have five favorable outcomes. So five is our numerator. Then our total number of possible outcomes is five as well. There are five total cards. So our denominator is five. And that's our fraction 5 over 5. 5 fths. We have a five out of five chance of picking an odd number. Now let's divide the numerator by the denominator to get our decimal. That's going to be one. One whole. Keep in mind whenever we have the same number on the top and bottom, our decimal is just one. Now since this is a whole number, we typically don't write the decimal point. But remember a decimal point comes after a whole number after the ones place. Then our percent here, we multiply that decimal by 100 and that's going to be 100%. Five out of five cards have an odd number, so we have a 100% chance. This is what we call a certain event. We're 100% certain it will happen. Let's move on to number three. What's the probability of picking a card with hearts? favorable outcomes. Well, there are three cards with hearts. So, we have three favorable outcomes. That's our numerator. And as far as the total number of possible outcomes, that's five. There are five total cards. So, our fraction is 3 over 5, 35ths. We have a three out of five chance of picking hearts. As far as our decimal here, 3 / 5 gives us 0.6 6/10. Then we can multiply that decimal by 100 to get our percent. Our percent here is 60%. And that's our probability of picking hearts. Moving on to number four, we have what's the probability of picking a queen? Favorable outcomes here. Well, it looks like we don't have any queens at all. So, this is going to be zero. We have zero favorable outcomes. There aren't any ways that this event can happen. And then our total number of possible outcomes is five. There are five cards in total. And that's our fraction 0 over five. 0 fifths. We have a zero out of five chance of picking a queen. Then as far as our decimal, that's going to be zero. And our percent is going to be 0%. We have no chance of picking a queen. This is what we call an impossible event. Lastly, let's move on to number five where we have what's the probability of picking a three or a five. So think if we randomly pick one card, we want either a three or a five. As far as favorable outcomes, how many cards have a three or a five? We have one three and one five. So that gives us two favorable outcomes over the total number of possible outcomes, five. Our fraction 2 over 5, two fifths. We have a two out of five chance of picking a three or a five. There are two cards that we can pick here that will make our event happen. Either will work. Picking the five card or picking the three card. Now for the decimal, 2 / 5 gives us 0.4 4/10. And then multiplying that decimal by 100 gives us our percent 40%. And that's our probability of picking a three or a five. So there's our second section of probability examples. Let's move on to the probability line. Here's our last section where we will take a look at the probability line. This is a nice conclusion to our intro to probability. This will build on what we've covered so far in this video. Remember, probability tells us how likely something is to happen. The probability line gives us a visual of the likelihood an event happens. We will go through five probability examples here. We will then place them on the probability line and talk about the likelihood of each event. We are going to use this spinner right here for our probability examples. So, if we spin the spinner, we're going to look at the probability of where it lands. You'll notice there are eight equal-sized sections on the spinner. So, each section has an equal chance of being landed on. Let's jump into number one where we have the probability of spinning a 10 for each example. Here I've already calculated the probability. So here there aren't any tens on the spinner. So we have zero favorable outcomes. There are eight total sections that the spinner can land on. So eight is our total number of possible outcomes. Then our decimal is zero and our percent is 0%. So we have our probability as a fraction decimal and percent. Now taking a look at this probability there's no chance we land on a 10. And if we go down to the probability line which is down here we can see that it starts at zero and goes up to one. All probabilities will fall somewhere on this line. And then we can see that we have 1/2 in the middle. We can use fractions, decimals, or percents when working with the probability line. So I have all three below. Now thinking about number one, a probability of zero falls right here. So I'm going to put a star and then I'll put a 10 next to it to label it. When we have a probability of zero, we call that an impossible event. There's no way for it to happen. So, for example, spinning a 10 here is an impossible event. There's no 10 on the spinner, so we can't spin a 10. Let's move on to number two, where we have the probability of landing on red or blue. So, either of those is a favorable outcome here. Either will work. As far as this event happening, there are four red sections and four blue sections on the spinner. 4 + 4 gives us a total of eight favorable outcomes. So, eight favorable outcomes over eight total possible outcomes. So, our fraction is 8 over 8. Our decimal is 1 and our percent is 100%. All eight sections are either red or blue. We have a 100% chance of landing on red or blue. This event is right here on the probability line. Our fraction is one. Remember whenever we have the same numerator and denominator, that fraction equals 1. And then our decimal is one and our percent is 100%. I'll write red. or blue here to label this. When we have a probability of one or 100%, we call that a certain event. We are 100% certain it will happen. Next, for number three, we have the probability of landing on an odd number. The odd numbers are 1, 3, 5, and seven. So that's four favorable outcomes over eight total possible outcomes. So there's a four out of eight chance of landing on an odd number. Now you'll notice I also have 1/2 written right here. That's because 4/8s simplifies to 1/2. Depending on the situation, class, the directions, whatever the case may be, you may need to simplify the fractions. So, I did want to include that. Something to keep in mind. Then we have our decimal 5/10 and our percent is 50%. This event is right here on the probability line. It's right in the middle. And I'll put odd in order to label it. This is what we call an evench chance event or a 5050 event. It's just as likely to happen as it is not to happen. Just like flipping a coin. 50% chance of heads, 50% chance of tails. For this example, we have a 50% chance of landing on an odd number and a 50% chance of landing on an even number. So again, this is an evench chance event. Moving on to number four, we have the probability of landing on a one or a two. So both are favorable outcomes. Either will work. The spinner has one section with a one and one section with a two. 1 + 1 gives us a total of two favorable outcomes over eight, which is the total number of possible outcomes. there's a 2 out of eight chance of spinning a 1 or a two. Now 2/8s simplifies to 1/4. So I wanted to include that as well. Our decimal here is 25 hundreds and our percent is 25%. Now this event is going to be right around here on the probability line. I'll put our fraction. So 28s which simplifies to 1/4 our decimal 25 hundreds and then our percent 25%. So I'll put those underneath and then let's label this one or two. This is what we call an unlikely event. It's more unlikely to happen than it is to happen. Yes, of course, unlikely events happen, but there's less than a 50% chance. There's a higher probability it doesn't happen. And let's think about this. There's two favorable outcomes out of eight total outcomes. One section with a one and one section with a two. So again, two favorable outcomes. And that's up against these six other sections that have other numbers. So think about what has a better chance of happening. Landing on a one or a two or landing on any of the other numbers. So again, this is an unlikely event. An event in between 0 and 1/2 is an unlikely event. Lastly, let's take a look at number five. We have the probability of landing on a number less than eight. So that's 1 2 3 four five six or seven. Those are all of the numbers less than eight on the spinner. That gives us seven favorable outcomes over eight total possible outcomes. So we have a 7 out of eight chance of landing on a number less than 8. Our decimal is 875,000 and our percent is 87.5%. This event is about right here on the probability line. So let's put the fraction decimal and percent. And then we can label this less than eight. This is what we call a likely event. It's more likely to happen than unlikely to happen. There's a higher probability it happens than doesn't happen. We have more than a 50% chance here. Anything in between 1/2 and 1 is a likely event. And thinking about this specific event here, we have seven favorable outcomes. Seven sections with a number less than eight. And that's up against only one section with a number not less than 8. So this is a likely event. Now, to wrap things up here as an overview of the probability line, the closer an event is to zero, the less likely it is to happen. So, the further left we go, the lower the probability. The closer an event is to one, the more likely it is to happen. So, the further right we go, the higher the probability. So, there you have it. There's an introduction to probability. I hope that helped. Thanks so much for watching. Until next time, peace.