Probability is about estimating or calculating how likely or probable something is to occur. The chance of a particular event transpiring can be described using words like "certain", "impossible" or "likely". In Maths, the probability is always displayed as either a fraction, decimal or percentage with values between 0 and 1. Any outcome deemed to be impossible will have a probability of 0 whilst an outcome that is certain will be 1. Probabilities cannot be larger than the value of 1.
Okay, guys, in this video we're going to be looking at probabilities. Probabilities are involved in all sorts of places in life, whether it's figuring out how to win your favourite gambling game or what's, whether it's figuring out Schrodinger's equation. In quantum mechanics, probabilities are ubiquitous throughout the natural world. They crop up in some places you wouldn't expect them as well. But for this video, we're just going to be looking at the basics.
And probabilities are simply decimals, a fraction or a percentage. They're all the same thing, really. In this case, we're giving it as decimals, zero. Five, for example, can be written as a half, or it could be written as 50% fractions, decimals or percentages. So we have a range from zero to one.
A probability cannot be larger than one, it can't be less than zero. A probability of zero means something is impossible. A probability of one means that it's absolutely certain and a probability of 0.5 means that there's an even chance it's halfway between. If you, for example, flipper heads and tails coin, there's an even chance in theory that it ends up as heads or tails. So it would be a 0.5 chance of a head.
And lower down on the scale, very unlikely, higher up, very likely. Probability is essentially measured as the number of ways that you come up with something divided by the total possible outcomes. So let's say we roll a dice, or roll a die rather, and we come up with the number three. What was the probability that we came up with the number three? Well, what are the number of ways in which we can make three?
Only one. There's only one side with a three, so it's that one three divided by the total possible outcomes, of which there are six sides to the dice. There are six possible ways that it could have gone. And so it's one over six or one divided by six or zero recurring. So the probability of rolling a three is one.
There's only one three available, divided by the total possible outcomes, six. Let's say we have a list of names instead. Let's take John. Ella, Dave and John. Again, what is the probability?
And probabilities are often written as P with brackets like this. What is the probability of in this list? We're going to choose something at random. What's the probability we land on John? Well, how many ways can we land on John?
There's one way and that's the first John, and then there's a second way that's the fourth John. So there are two ways to land on John divided by the total number of possible outcomes. That's 1234 in the list. So we have the probability is a half or 0.5. There's an even chance that we will end up on John or not John.
Finally, let's do one last one. What's the probability of not Ella? What's the probability that I do not land on Ella? Well, how many ways are there that I do not land on Ella? There are one, two, three ways.
There are one, two, three ways that I don't land on Ella. And there are four possible outcomes. So you've got three over four, or zero point 75 rather chance 75% chance of not landing on Ella.
An experimental probability, also called relative frequency, is different from the case of a dice. If you have a fair dice and you roll a three, you've got a one in six chance. That's a theoretical probability. An experimental probability is done with testing, with experiments. So Ben throws a die ten times and he rolls four threes.
What's the relative frequency or the experimental probability that Ben rolls a three? We don't have to consider the theoretical case. All we do is we look at what was done in this experiment. So we had four threes divided by so it was four threes out of ten rolls. And so we get 0.4 for our probability that Ben rolls a three.
In this experimental probability, what do you think will happen if the die is fair and Ben rolls 100 times rather than ten times? We already saw that the chance in a fair die is one in six or zero point 16 recurring, which is less than this. So what do you think is going to happen to Ben's experiment if he rolls the die 100 times? Well, that zero four is going to go down and down and down and it's going to start approaching the true theoretical probability. Assuming it's a fair die, how many ways are there to order the letters A, B and C?
This is a common question in your exams. It's just basically listing these different ways in which it can occur. The trick is to come up with some systematic approach. So here we're going to do it in alphabetical order wherever possible. So we're going to start with listing the obvious one, A-B-C and there's another A.
We're going to do it in alphabetical order where possible. So we'll start with a we can't do BC, we've already done that, but we can do CB. Can we start another one with A? No, because we've already got those. So now we'll go on alphabetical loader, where possible, B, and again alphabetical loaded where possible.
We'll start with A, then C. We can still start with a B, but now it's got to be a C and then followed by an A. And we've done all the raise, we've done all our B's. So now, starting with a C, again alphabetical order where possible, a comes first, followed by B, now is C and we can't do A. So it's BA C-B-A.
So how many ways are there to order the letters? There are six ways to order.
Sample spaces are very useful and you'll see why. Right now, here's a spinner and it's a fair spinner with one, two and three an equal probability of landing on one, two or three. I'm going to spin this spinner twice and I'm going to add up those two results. So if I span a two and a three, I'd add them up to get a five. What is the most likely outcome?
Now you might be thinking, well, what difference does it make? If it's a fair spinner, I'm just as likely to get a one plus three, which is four, than I am to get a two plus three, which is five. So it doesn't matter, all my outcomes are equally likely, right? I'm afraid that's not true. You'll see why right now I'm going to draw out change the colour of that.
You'll see why right now can draw a sample space table here. And what we're going to do here is we're going to list the different results that we could get on the spinner vertically and horizontally. And we're going to assume that we get these and then add them up. So this is at the top, throw one and this is along the left, throw two. So throw one, I get a one.
Throw two, I get a one, I add them up. One plus one is two. Throw one, I get a one. Again, throw two, I get a two, add them up and get three and so on. So I add up all the possibilities to here.
Two plus one is three, two plus two is four, two plus three is five. Throw one is a three, throw two is a one, throw one is a three, throw two is a two, and throw one is a three, and throw two is a three. These are all the possibilities. What is the most likely outcome? Well, notice here are all our nine possible outcomes.
There are three ways that I can get a four, I can get a three four by one. I can get a two four, x two or a one four by three. There is only one way I can get a two that's a one and a one and only one way I can get a six, which is a three and a three. So as you can see, a sample space shows us very clearly that the outcomes are not equally likely. So the answer is the most likely outcome is four.
Time for an exam question. There are 26 suites in a bag. 15 of them are red, the rest are white. One of the suites is taken at random. Find the probability that the suite is red.
Okay, nice and straightforward. This. 126 suites in the bag in total. So we know it's going to be out of 26. We just need to know how many reds there are.
There are 15 reds. Are any of the other suites going to be red? No, it tells us very clearly the rest are white. So we've got 15 chances to get a red out of a total of 26 chances so the final answer is 15 over 26. Now, you can turn this into a decimal, or if it's an unsimplified fraction, you can simplify it.
But that is a good enough answer for an exam question. Just putting the final probability in the answer as a fraction or a decimal.
Okay, time for one for you all. Let's see how you get on with this. We are dealing with a biassed spinner and see if you can first complete the table and then work out the estimate for the number of times the spinner will land on two, given that Johnny throws it, spins it 200 times. I'd like to see your answers in the comments section. If you would like to see if you're right, let me know.
My name is Adam. A Tudor for Sherpa. Any other tutor, for sure. But I can also give you the answer to this, but please do have fun trying to get this one. I'll see you in the comments section.
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