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Hi I’m Rob, welcome to Math Antics!
In this video we’re gonna learn about how to do math with things that only sometimes happen.
They might be likely or unlikely.
We’re gonna learn about Probability.
Usually in math we deal with things that always happen the same way.
They’re completely certain.
Like if you add 1 and 1 you’re always gonna get 2.
If you multiply 2 and 3 you’re always gonna get 6.
There’s no uncertainty at all.
But in the real world things aren’t always so predictable.
Take a coin toss for example.
We can’t predict whether it will be heads or tails.
It’s unpredictable or random, and that’s why some people will
flip a coin to help decide which of two things to do.
That’s how I make every decision in life
Why am I not surprised?
[thunder strike ]
But even though we don’t know what each coin flip is going to be,
we do know a few things about it.
We know that with a fair coin toss that heads is just as likely to show up as tails.
The Probability of an event (like getting heads or getting tails)
is a value that tells us HOW LIKELY that event is to happen.
With our coin toss, since each side is just as likely,
and there’s only two sides to a coin,
if we flipped a coin a lot of times,
we should expect that about half the flips will be heads
and about half the flips will be tails.
And that means the probability of flipping heads is the fraction one-half
and the probability of flipping tails is also one-half.
Let’s look at this in a little more detail on something called a Probability Line.
It’s a number line that goes from 0 to 1.
A probability of zero means that an event cannot happen, it’s impossible.
And a probability of 1 means that an event is definitely going to happen, it’s certain.
That’s why the probability line only goes from 0 to 1.
An event can’t be less likely than impossible
and it can’t be more likely than certain.
A probability of one-half (like with our coin toss)
means that an event is just as likely to happen as it is to NOT happen.
A probability less than one-half means that an event is unlikely
and a probability greater than one-half means that an event is likely.
Oh, and in addition to fractions,
it’s also common to write probabilities as decimals or percentages,
since you can easily convert between those three.
A probability of 0 is the same as a 0 percent chance of something happening,
a probability of one-half is the same as a 50 percent chance of something happening,
and a probability of 1 is the same as a 100 percent chance of something happening.
Now that you know how a coin toss works,
let’s see an example of an event that is unlikely using something a little more complicated than a coin.
Let’s take a look at dice.
A standard die has 6 sides numbered 1 through 6.
When you roll it, any of those sides is just as likely to come up as the others.
That sounds a lot like flipping a coin doesn’t it?
Each side of a die is just as likely to come up as the others
and each side of a coin was just as likely to come up as the other.
So you might expect that the probability of rolling a 3 is 50%.
But remember, with a coin toss there were only 2 possibilities: heads or tails.
With dice there are 6 possibilities.
And that’s going to make a difference in its probability.
One way to think about it is that it’s certain that one of those six sides will land facing upwards,
which is a probability of 1 (or 100%)
But since ONLY one side can face upwards for a given roll,
we have to divide up that value among all the possibilities.
In the case of a coin toss, since there were only 2 possibilities,
we had to divide the probability by 2.
1 divided by 2 is one-half, (which is the decimal 0.5 or 50%)
But with the die, we need to divide the probability up evenly between 6 possibilities.
1 divided by 6 is one-sixth which is equivalent to 0.167 (or 16.7%)
So that would be right here on our probability line.
That means it isn’t likely that I would roll a 3 for instance,
but it’s just as likely as rolling any other number.
And since all 6 numbers have the same probability,
each number should come up about as often as the others.
To see if they do, I’m going to conduct some trials.
That’s an excellent argument.
Allow me to deliberate.
Actually, when dealing with probability,
a trial (which can also be called an experiment)
is a process that has a random outcome
…like tossing a coin or rolling dice or spinning a spinner.
And the outcome of a trial is what happens in that particular trial.
Like flipping heads, or rolling a 3.
So I’m gonna conduct several trials by rolling a die multiple times
and keeping track of how many times I roll each number.
Ah ha! You said that each number would come up just as often as the other numbers.
But look! There’s more 2s here than there are 5s.
How do you explain that?!
Well remember, we’re dealing with things that are random. They’re unpredictable.
We can’t know exactly what will happen, just what will happen on average.
So now I have to calculate the average?
Well, when we say “on average” we mean that
the more trials you do the closer you get to the expected probabilities.
There. Now that we’ve done a LOT of trials
you can see that our totals are much closer to what you would expect them to be.
I guess you’re probably right.
That’s one of the really important things to keep in mind about probability.
If you do just a few trials, the results might not end up very close to what you’d expect.
In fact, they could be way off!
But if you do more trials, you increase your chances of reaching the expected probabilities.
There’s another thing I should point out.
Remember, the probability of flipping heads is 1/2 and the probability of flipping tails is 1/2.
The probability of rolling a 1 is 1/6, and the probability of rolling any other number on a die is 1/6.
If you add up the probabilities for the coin flip, you get 2/2 or 1.
And if you add up the probabilities for rolling a die, you get 6/6 which is also 1.
And that’s not just a coincidence.
If you add up the probabilities of all possible outcomes of a trial,
the total is going to be 1 or 100%
because it is certain that at least one of those possibilities will happen.
Let’s look at some more examples.
For these examples we’ll use a spinner.
If we had a spinner with just six equally sized sectors,
the probabilities would be exactly the same as with dice.
So we want a few more sectors.
There, that’s more like it. Now we have 16 equally sized sectors.
So, what is the probability of spinning a 12?
Well, just like with the dice where, we had to split up the 100% between all six possibilities,
we’ll do the same thing now, but we’ll split it up between 16 possibilities.
So the probability of spinning a 12 is 1/16 or about 6%
which is right here on the probability line.
We can see that the probability of spinning a 12 is less likely than the probability of rolling a 3.
And that makes sense because there are more possible outcomes with our spinner.
But what if we color some of the sectors a different color
and we want to know the probability of spinning a certain color?
Now we have 5 sectors colored blue and 11 sectors colored yellow.
So what is the probability of spinning a blue?
Remember how with a coin toss, we ended up with the fraction 1 over 2
and with a die roll we got the fraction 1 over 6.
In both cases we had 1 as the numerator.
And that’s because we were interested in only ONE of the possible outcomes,
like the probability of flipping heads or the probability of the number 3 being rolled.
But in this case the top number of our fraction will be 5
because any of these 5 sectors will give us the color we want.
And the bottom number will still be the total number of possibilities,
which is 16 because that’s how many total sectors we have.
So the probability of spinning a blue is 5/16 or about 31%.
That’s still considered unlikely,
but it is more likely than spinning a specific number.
And this method will work for figuring out the probability of any event.
You just make a fraction with the numerator as the number of outcomes that satisfy your requirement,
and the denominator as the total number of possible outcomes.
Let’s try the same method to find the probability of spinning a yellow.
Our top number should be 11 because there’s 11 yellow sectors.
And our bottom number should still be 16. So the probability of spinning a yellow is 11/16 or about 69%.
Now we finally have a probability that’s considered likely.
And it makes sense, because you can see by looking at our spinner
that it’s more likely to spin a yellow than a blue.
And you’ll notice, if we add up 5/16 and 11/16 we get 16/16 or a probability of 1.
So that’s a good sign that we did it right.
Let’s look at another example. Suppose we have a bag of marbles.
There are 3 green marbles, 7 yellow marbles and 1 white marble.
If we mix them up and pull out a marble at random,
what’s the probability of it being green?
Well, the top number of our probability fraction will be 3
because there’s 3 green marbles so there’s 3 outcomes that get us what we want.
And the bottom number will be 11 because there’s a total of 11 possible marbles that we could pull out.
So the probability of pulling out a green marble is 3/11 or 0.27 or 27%
It’s right here on the probability line. That means it’s unlikely.
And that makes sense because you can see that
it would be less likely to pull out a green marble than one of the other ones.
Let’s try this again for calculating the probability of pulling out a yellow marble.
This time the numerator of our fraction will be 7 because there’s 7 yellow marbles.
The denominator will still be 11 because there are still 11 marbles total.
So the probability of pulling out a yellow marble is 7/11 or 0.64 or 64%.
…another example of an event that is likely.
…how about pulling out the white marble?
Well, the top number will be 1 since there’s only one white marble.
And the bottom number is still 11.
So the probability of pulling out a white marble is 1/11 or 0.09 or 9%
…not very likely.
And if we add up these probabilities we get 11/11, or 100%, just as we expected.
Alright! So you should have a pretty good handle on basic probability now.
You just have to remember to make a fraction with
the numerator being the number of outcomes that give you what you want,
and the denominator being the total number of possibilities.
And we learned about the Probability Line,
and that a probability can’t be less than 0 or greater that 1 (or 100%).
We also learned that the more trials or experiments you conduct,
the closer your results will get to the expected probabilities.
Of course the way to get good at it is to practice.
So be sure to do a lot of problems on your own.
As always, thanks for watching Math Antics and I’ll see ya next time.
And I sentence you to…
…five years hard labor!
Learn more at www.mathantics.com